├── ProblemSet1.pdf ├── README.md ├── problem_set_1_solutions.py ├── ProblemSet1_Solutions.ipynb ├── ComplexConjugate.ipynb ├── VisualizingComplexNumbers.nb ├── ProblemSet1.nb └── DerivingThePolarFormOfComplexNumbers.nb /ProblemSet1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/juanklopper/ComplexAnalysis/main/ProblemSet1.pdf -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # ComplexAnalysis 2 | Various files for my course on an introduction to complex analysis. The files are Mathematica notebooks. 3 | The video lectures are available at https://www.youtube.com/playlist?list=PLsu0TcgLDUiJ00m2zweg7HNG1QWGlP2d9 4 | -------------------------------------------------------------------------------- /problem_set_1_solutions.py: -------------------------------------------------------------------------------- 1 | """ 2 | Problem Set 1 Solutions 3 | Complex Analysis Problems using NumPy and SciPy 4 | """ 5 | 6 | import numpy as np 7 | from scipy import linalg 8 | 9 | def print_problem(num, description, result): 10 | """Helper function to print problem results""" 11 | print(f"\n{'='*60}") 12 | print(f"Problem {num}") 13 | print(f"{description}") 14 | print(f"Result: {result}") 15 | print(f"{'='*60}") 16 | 17 | def complex_to_polar(z): 18 | """Convert complex number to polar form (r, theta)""" 19 | r = np.abs(z) 20 | theta = np.angle(z) 21 | return r, theta 22 | 23 | def polar_to_complex(r, theta): 24 | """Convert polar form to complex number""" 25 | return r * np.exp(1j * theta) 26 | 27 | # Problem 1: (3 + 2i) + (-7 - i) 28 | z1 = (3 + 2j) + (-7 - 1j) 29 | print_problem(1, "(3 + 2i) + (-7 - i)", z1) 30 | 31 | # Problem 2: [(5 + 3i) + (-1 + 2i)] - (7 - 5i) 32 | z2 = ((5 + 3j) + (-1 + 2j)) - (7 - 5j) 33 | print_problem(2, "[(5 + 3i) + (-1 + 2i)] - (7 - 5i)", z2) 34 | 35 | # Problem 3: (5 + 3i) + [(-1 + 2i) - (7 - 5i)] 36 | z3 = (5 + 3j) + ((-1 + 2j) - (7 - 5j)) 37 | print_problem(3, "(5 + 3i) + [(-1 + 2i) - (7 - 5i)]", z3) 38 | 39 | # Problem 4: (2 - 3i)(4 + 2i) 40 | z4 = (2 - 3j) * (4 + 2j) 41 | print_problem(4, "(2 - 3i)(4 + 2i)", z4) 42 | 43 | # Problem 5: (4 + 2i)(2 - 3i) 44 | z5 = (4 + 2j) * (2 - 3j) 45 | print_problem(5, "(4 + 2i)(2 - 3i)", z5) 46 | 47 | # Problem 6: (2 - i)[(-3 + 2i)(5 - 4i)] 48 | z6 = (2 - 1j) * ((-3 + 2j) * (5 - 4j)) 49 | print_problem(6, "(2 - i)[(-3 + 2i)(5 - 4i)]", z6) 50 | 51 | # Problem 7: [(2 - i)(-3 + 2i)](5 - 4i) 52 | z7 = ((2 - 1j) * (-3 + 2j)) * (5 - 4j) 53 | print_problem(7, "[(2 - i)(-3 + 2i)](5 - 4i)", z7) 54 | 55 | # Problem 8: (-1 + 2i)[(7 - 5i) + (-3 + 4i)] 56 | z8 = (-1 + 2j) * ((7 - 5j) + (-3 + 4j)) 57 | print_problem(8, "(-1 + 2i)[(7 - 5i) + (-3 + 4i)]", z8) 58 | 59 | # Problem 9: Let z1 = 2 + i, z2 = 3 - 2i. Calculate |3z1 - 4z2| 60 | z1_p9 = 2 + 1j 61 | z2_p9 = 3 - 2j 62 | result_p9 = 3 * z1_p9 - 4 * z2_p9 63 | magnitude_p9 = np.abs(result_p9) 64 | print_problem(9, "Let z1 = 2 + i, z2 = 3 - 2i. Calculate |3z1 - 4z2|", 65 | f"{result_p9}, |3z1 - 4z2| = {magnitude_p9}") 66 | 67 | # Problem 10: Let z = 2 + i. Calculate z³ - 3z² + 4z - 8 68 | z_p10 = 2 + 1j 69 | result_p10 = z_p10**3 - 3*z_p10**2 + 4*z_p10 - 8 70 | print_problem(10, "Let z = 2 + i. Calculate z³ - 3z² + 4z - 8", result_p10) 71 | 72 | # Problem 11: Let z = -1/2 + (√3/2)i. Calculate z⁴ 73 | z_p11 = -1/2 + (np.sqrt(3)/2)*1j 74 | result_p11 = z_p11**4 75 | print_problem(11, "Let z = -1/2 + (√3/2)i. Calculate z⁴", result_p11) 76 | 77 | # Problem 12: Find x and y if 3x + 2yi - ix + 5y = 7 + 5i 78 | # Rearranging: (3x + 5y) + i(-x + 2y) = 7 + 5i 79 | # This gives us two equations: 80 | # 3x + 5y = 7 81 | # -x + 2y = 5 82 | A = np.array([[3, 5], [-1, 2]]) 83 | b = np.array([7, 5]) 84 | solution_p12 = linalg.solve(A, b) 85 | x_p12, y_p12 = solution_p12 86 | print_problem(12, "Find x and y if 3x + 2yi - ix + 5y = 7 + 5i", 87 | f"x = {x_p12}, y = {y_p12}") 88 | 89 | # Problem 13: Express 2 + 2√3 i in polar form 90 | z_p13 = 2 + 2*np.sqrt(3)*1j 91 | r_p13, theta_p13 = complex_to_polar(z_p13) 92 | print_problem(13, "Express 2 + 2√3 i in polar form", 93 | f"r = {r_p13}, θ = {theta_p13} rad = {theta_p13/np.pi}π rad\n" + 94 | f"Polar form: {r_p13}e^(i·{theta_p13/np.pi}π)") 95 | 96 | # Problem 14: Express -5 + 5i in polar form 97 | z_p14 = -5 + 5j 98 | r_p14, theta_p14 = complex_to_polar(z_p14) 99 | print_problem(14, "Express -5 + 5i in polar form", 100 | f"r = {r_p14:.6f}, θ = {theta_p14} rad = {theta_p14/np.pi}π rad\n" + 101 | f"Polar form: {r_p14:.6f}e^(i·{theta_p14/np.pi}π)") 102 | 103 | # Problem 15: Express 3e^(πi/2) in rectangular form 104 | r_p15 = 3 105 | theta_p15 = np.pi / 2 106 | z_p15 = polar_to_complex(r_p15, theta_p15) 107 | print_problem(15, "Express 3e^(πi/2) in rectangular form", 108 | f"{z_p15}\n" + 109 | f"Real part: {z_p15.real:.10f} ≈ 0\n" + 110 | f"Imaginary part: {z_p15.imag:.10f} = 3") 111 | 112 | # Summary 113 | print("\n" + "="*60) 114 | print("SUMMARY OF ALL SOLUTIONS") 115 | print("="*60) 116 | solutions = { 117 | "Problem 1": z1, 118 | "Problem 2": z2, 119 | "Problem 3": z3, 120 | "Problem 4": z4, 121 | "Problem 5": z5, 122 | "Problem 6": z6, 123 | "Problem 7": z7, 124 | "Problem 8": z8, 125 | "Problem 9": f"|3z1 - 4z2| = {magnitude_p9}", 126 | "Problem 10": result_p10, 127 | "Problem 11": result_p11, 128 | "Problem 12": f"x = {x_p12}, y = {y_p12}", 129 | "Problem 13": f"{r_p13}e^(i·π/3)", 130 | "Problem 14": f"{r_p14:.6f}e^(i·3π/4)", 131 | "Problem 15": "0 + 3i" 132 | } 133 | 134 | for problem, solution in solutions.items(): 135 | print(f"{problem}: {solution}") 136 | print("="*60) 137 | -------------------------------------------------------------------------------- /ProblemSet1_Solutions.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "# Problem Set 1 - Complex Analysis Solutions\n", 8 | "\n", 9 | "Solutions to complex analysis problems using Python with NumPy and SciPy." 10 | ] 11 | }, 12 | { 13 | "cell_type": "code", 14 | "execution_count": null, 15 | "metadata": {}, 16 | "outputs": [], 17 | "source": [ 18 | "import numpy as np\n", 19 | "from scipy import linalg\n", 20 | "import matplotlib.pyplot as plt" 21 | ] 22 | }, 23 | { 24 | "cell_type": "markdown", 25 | "metadata": {}, 26 | "source": [ 27 | "## Helper Functions" 28 | ] 29 | }, 30 | { 31 | "cell_type": "markdown", 32 | "metadata": {}, 33 | "source": [ 34 | "$$\n", 35 | "\\begin{align*}\n", 36 | "&r = \\lvert z \\rvert = \\sqrt{x^{2} + y^{2}} \\\\ \\\\\n", 37 | "&\\theta = \\tan^{-1}\\left( \\frac{y}{x} \\right) \\\\ \\\\\n", 38 | "&z = r e^{i \\theta}\n", 39 | "\\end{align*}\n", 40 | "$$\n" 41 | ] 42 | }, 43 | { 44 | "cell_type": "code", 45 | "execution_count": null, 46 | "metadata": {}, 47 | "outputs": [], 48 | "source": [ 49 | "def complex_to_polar(z):\n", 50 | " \"\"\"Convert complex number to polar form (r, theta)\"\"\"\n", 51 | " r = np.abs(z)\n", 52 | " theta = np.angle(z)\n", 53 | " return r, theta\n", 54 | "\n", 55 | "def polar_to_complex(r, theta):\n", 56 | " \"\"\"Convert polar form to complex number\"\"\"\n", 57 | " return r * np.exp(1j * theta)" 58 | ] 59 | }, 60 | { 61 | "cell_type": "markdown", 62 | "metadata": {}, 63 | "source": [ 64 | "## Problem 1\n", 65 | "Calculate: $(3 + 2i) + (-7 - i)$" 66 | ] 67 | }, 68 | { 69 | "cell_type": "code", 70 | "execution_count": null, 71 | "metadata": {}, 72 | "outputs": [], 73 | "source": [ 74 | "z1 = (3 + 2j) + (-7 - 1j)\n", 75 | "print(f\"Problem 1: (3 + 2i) + (-7 - i) = {z1}\")" 76 | ] 77 | }, 78 | { 79 | "cell_type": "markdown", 80 | "metadata": {}, 81 | "source": [ 82 | "## Problem 2\n", 83 | "Calculate: $[(5 + 3i) + (-1 + 2i)] - (7 - 5i)$" 84 | ] 85 | }, 86 | { 87 | "cell_type": "code", 88 | "execution_count": null, 89 | "metadata": {}, 90 | "outputs": [], 91 | "source": [ 92 | "z2 = ((5 + 3j) + (-1 + 2j)) - (7 - 5j)\n", 93 | "print(f\"Problem 2: [(5 + 3i) + (-1 + 2i)] - (7 - 5i) = {z2}\")" 94 | ] 95 | }, 96 | { 97 | "cell_type": "markdown", 98 | "metadata": {}, 99 | "source": [ 100 | "## Problem 3\n", 101 | "Calculate: $(5 + 3i) + [(-1 + 2i) - (7 - 5i)]$" 102 | ] 103 | }, 104 | { 105 | "cell_type": "code", 106 | "execution_count": null, 107 | "metadata": {}, 108 | "outputs": [], 109 | "source": [ 110 | "z3 = (5 + 3j) + ((-1 + 2j) - (7 - 5j))\n", 111 | "print(f\"Problem 3: (5 + 3i) + [(-1 + 2i) - (7 - 5i)] = {z3}\")" 112 | ] 113 | }, 114 | { 115 | "cell_type": "markdown", 116 | "metadata": {}, 117 | "source": [ 118 | "## Problem 4\n", 119 | "Calculate: $(2 - 3i)(4 + 2i)$" 120 | ] 121 | }, 122 | { 123 | "cell_type": "code", 124 | "execution_count": null, 125 | "metadata": {}, 126 | "outputs": [], 127 | "source": [ 128 | "z4 = (2 - 3j) * (4 + 2j)\n", 129 | "print(f\"Problem 4: (2 - 3i)(4 + 2i) = {z4}\")" 130 | ] 131 | }, 132 | { 133 | "cell_type": "markdown", 134 | "metadata": {}, 135 | "source": [ 136 | "## Problem 5\n", 137 | "Calculate: $(4 + 2i)(2 - 3i)$" 138 | ] 139 | }, 140 | { 141 | "cell_type": "code", 142 | "execution_count": null, 143 | "metadata": {}, 144 | "outputs": [], 145 | "source": [ 146 | "z5 = (4 + 2j) * (2 - 3j)\n", 147 | "print(f\"Problem 5: (4 + 2i)(2 - 3i) = {z5}\")" 148 | ] 149 | }, 150 | { 151 | "cell_type": "markdown", 152 | "metadata": {}, 153 | "source": [ 154 | "## Problem 6\n", 155 | "Calculate: $(2 - i)[(-3 + 2i)(5 - 4i)]$" 156 | ] 157 | }, 158 | { 159 | "cell_type": "code", 160 | "execution_count": null, 161 | "metadata": {}, 162 | "outputs": [], 163 | "source": [ 164 | "z6 = (2 - 1j) * ((-3 + 2j) * (5 - 4j))\n", 165 | "print(f\"Problem 6: (2 - i)[(-3 + 2i)(5 - 4i)] = {z6}\")" 166 | ] 167 | }, 168 | { 169 | "cell_type": "markdown", 170 | "metadata": {}, 171 | "source": [ 172 | "## Problem 7\n", 173 | "Calculate: $[(2 - i)(-3 + 2i)](5 - 4i)$" 174 | ] 175 | }, 176 | { 177 | "cell_type": "code", 178 | "execution_count": null, 179 | "metadata": {}, 180 | "outputs": [], 181 | "source": [ 182 | "z7 = ((2 - 1j) * (-3 + 2j)) * (5 - 4j)\n", 183 | "print(f\"Problem 7: [(2 - i)(-3 + 2i)](5 - 4i) = {z7}\")" 184 | ] 185 | }, 186 | { 187 | "cell_type": "markdown", 188 | "metadata": {}, 189 | "source": [ 190 | "## Problem 8\n", 191 | "Calculate: $(-1 + 2i)[(7 - 5i) + (-3 + 4i)]$" 192 | ] 193 | }, 194 | { 195 | "cell_type": "code", 196 | "execution_count": null, 197 | "metadata": {}, 198 | "outputs": [], 199 | "source": [ 200 | "z8 = (-1 + 2j) * ((7 - 5j) + (-3 + 4j))\n", 201 | "print(f\"Problem 8: (-1 + 2i)[(7 - 5i) + (-3 + 4i)] = {z8}\")" 202 | ] 203 | }, 204 | { 205 | "cell_type": "markdown", 206 | "metadata": {}, 207 | "source": [ 208 | "## Problem 9\n", 209 | "Let $z_1 = 2 + i$ and $z_2 = 3 - 2i$. Calculate $|3z_1 - 4z_2|$" 210 | ] 211 | }, 212 | { 213 | "cell_type": "code", 214 | "execution_count": null, 215 | "metadata": {}, 216 | "outputs": [], 217 | "source": [ 218 | "z1_p9 = 2 + 1j\n", 219 | "z2_p9 = 3 - 2j\n", 220 | "result_p9 = 3 * z1_p9 - 4 * z2_p9\n", 221 | "magnitude_p9 = np.abs(result_p9)\n", 222 | "\n", 223 | "print(f\"z1 = {z1_p9}\")\n", 224 | "print(f\"z2 = {z2_p9}\")\n", 225 | "print(f\"3z1 - 4z2 = {result_p9}\")\n", 226 | "print(f\"|3z1 - 4z2| = {magnitude_p9}\")" 227 | ] 228 | }, 229 | { 230 | "cell_type": "markdown", 231 | "metadata": {}, 232 | "source": [ 233 | "## Problem 10\n", 234 | "Let $z = 2 + i$. Calculate $z^3 - 3z^2 + 4z - 8$" 235 | ] 236 | }, 237 | { 238 | "cell_type": "code", 239 | "execution_count": null, 240 | "metadata": {}, 241 | "outputs": [], 242 | "source": [ 243 | "z_p10 = 2 + 1j\n", 244 | "result_p10 = z_p10**3 - 3*z_p10**2 + 4*z_p10 - 8\n", 245 | "\n", 246 | "print(f\"z = {z_p10}\")\n", 247 | "print(f\"z³ - 3z² + 4z - 8 = {result_p10}\")" 248 | ] 249 | }, 250 | { 251 | "cell_type": "markdown", 252 | "metadata": {}, 253 | "source": [ 254 | "## Problem 11\n", 255 | "Let $z = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i$. Calculate $z^4$" 256 | ] 257 | }, 258 | { 259 | "cell_type": "code", 260 | "execution_count": null, 261 | "metadata": {}, 262 | "outputs": [], 263 | "source": [ 264 | "z_p11 = -1/2 + (np.sqrt(3)/2)*1j\n", 265 | "result_p11 = z_p11**4\n", 266 | "\n", 267 | "print(f\"z = {z_p11}\")\n", 268 | "print(f\"z⁴ = {result_p11}\")\n", 269 | "print(f\"z⁴ (real part) = {result_p11.real}\")\n", 270 | "print(f\"z⁴ (imaginary part) = {result_p11.imag}\")" 271 | ] 272 | }, 273 | { 274 | "cell_type": "markdown", 275 | "metadata": {}, 276 | "source": [ 277 | "## Problem 12\n", 278 | "Find $x$ and $y$ if $3x + 2yi - ix + 5y = 7 + 5i$\n", 279 | "\n", 280 | "Rearranging: $(3x + 5y) + i(-x + 2y) = 7 + 5i$\n", 281 | "\n", 282 | "This gives us two equations:\n", 283 | "- $3x + 5y = 7$\n", 284 | "- $-x + 2y = 5$" 285 | ] 286 | }, 287 | { 288 | "cell_type": "code", 289 | "execution_count": null, 290 | "metadata": {}, 291 | "outputs": [], 292 | "source": [ 293 | "# System of linear equations\n", 294 | "A = np.array([[3, 5], [-1, 2]])\n", 295 | "b = np.array([7, 5])\n", 296 | "solution_p12 = linalg.solve(A, b)\n", 297 | "x_p12, y_p12 = solution_p12\n", 298 | "\n", 299 | "print(f\"System of equations:\")\n", 300 | "print(f\" 3x + 5y = 7\")\n", 301 | "print(f\" -x + 2y = 5\")\n", 302 | "print(f\"\\nSolution:\")\n", 303 | "print(f\" x = {x_p12}\")\n", 304 | "print(f\" y = {y_p12}\")\n", 305 | "\n", 306 | "# Verify\n", 307 | "verification = 3*x_p12 + 2*y_p12*1j - 1j*x_p12 + 5*y_p12\n", 308 | "print(f\"\\nVerification: 3x + 2yi - ix + 5y = {verification}\")" 309 | ] 310 | }, 311 | { 312 | "cell_type": "markdown", 313 | "metadata": {}, 314 | "source": [ 315 | "## Problem 13\n", 316 | "Express $2 + 2\\sqrt{3}i$ in polar form" 317 | ] 318 | }, 319 | { 320 | "cell_type": "code", 321 | "execution_count": null, 322 | "metadata": {}, 323 | "outputs": [], 324 | "source": [ 325 | "z_p13 = 2 + 2*np.sqrt(3)*1j\n", 326 | "r_p13, theta_p13 = complex_to_polar(z_p13)\n", 327 | "\n", 328 | "print(f\"z = {z_p13}\")\n", 329 | "print(f\"r = |z| = {r_p13}\")\n", 330 | "print(f\"θ = arg(z) = {theta_p13} rad\")\n", 331 | "print(f\"θ = {theta_p13/np.pi}π rad\")\n", 332 | "print(f\"\\nPolar form: {r_p13}e^(i·π/3)\")\n", 333 | "print(f\"or: {r_p13}∠{np.degrees(theta_p13)}°\")" 334 | ] 335 | }, 336 | { 337 | "cell_type": "markdown", 338 | "metadata": {}, 339 | "source": [ 340 | "## Problem 14\n", 341 | "Express $-5 + 5i$ in polar form" 342 | ] 343 | }, 344 | { 345 | "cell_type": "code", 346 | "execution_count": null, 347 | "metadata": {}, 348 | "outputs": [], 349 | "source": [ 350 | "z_p14 = -5 + 5j\n", 351 | "r_p14, theta_p14 = complex_to_polar(z_p14)\n", 352 | "\n", 353 | "print(f\"z = {z_p14}\")\n", 354 | "print(f\"r = |z| = {r_p14}\")\n", 355 | "print(f\"θ = arg(z) = {theta_p14} rad\")\n", 356 | "print(f\"θ = {theta_p14/np.pi}π rad\")\n", 357 | "print(f\"\\nPolar form: {r_p14:.6f}e^(i·3π/4)\")\n", 358 | "print(f\"or: {r_p14:.6f}∠{np.degrees(theta_p14)}°\")" 359 | ] 360 | }, 361 | { 362 | "cell_type": "markdown", 363 | "metadata": {}, 364 | "source": [ 365 | "## Problem 15\n", 366 | "Express $3e^{\\frac{\\pi}{2}i}$ in rectangular form" 367 | ] 368 | }, 369 | { 370 | "cell_type": "code", 371 | "execution_count": null, 372 | "metadata": {}, 373 | "outputs": [], 374 | "source": [ 375 | "r_p15 = 3\n", 376 | "theta_p15 = np.pi / 2\n", 377 | "z_p15 = polar_to_complex(r_p15, theta_p15)\n", 378 | "\n", 379 | "print(f\"r = {r_p15}\")\n", 380 | "print(f\"θ = π/2 = {theta_p15} rad\")\n", 381 | "print(f\"\\nz = {z_p15}\")\n", 382 | "print(f\"\\nRectangular form: {z_p15.real:.10f} + {z_p15.imag}i\")\n", 383 | "print(f\"Simplified: 0 + 3i = 3i\")" 384 | ] 385 | }, 386 | { 387 | "cell_type": "markdown", 388 | "metadata": {}, 389 | "source": [ 390 | "## Summary of All Solutions" 391 | ] 392 | }, 393 | { 394 | "cell_type": "code", 395 | "execution_count": null, 396 | "metadata": {}, 397 | "outputs": [], 398 | "source": [ 399 | "print(\"=\"*60)\n", 400 | "print(\"SUMMARY OF ALL SOLUTIONS\")\n", 401 | "print(\"=\"*60)\n", 402 | "\n", 403 | "solutions = {\n", 404 | " \"Problem 1\": z1,\n", 405 | " \"Problem 2\": z2,\n", 406 | " \"Problem 3\": z3,\n", 407 | " \"Problem 4\": z4,\n", 408 | " \"Problem 5\": z5,\n", 409 | " \"Problem 6\": z6,\n", 410 | " \"Problem 7\": z7,\n", 411 | " \"Problem 8\": z8,\n", 412 | " \"Problem 9\": f\"|3z1 - 4z2| = {magnitude_p9}\",\n", 413 | " \"Problem 10\": result_p10,\n", 414 | " \"Problem 11\": result_p11,\n", 415 | " \"Problem 12\": f\"x = {x_p12}, y = {y_p12}\",\n", 416 | " \"Problem 13\": f\"{r_p13}e^(i·π/3)\",\n", 417 | " \"Problem 14\": f\"{r_p14:.6f}e^(i·3π/4)\",\n", 418 | " \"Problem 15\": \"0 + 3i\"\n", 419 | "}\n", 420 | "\n", 421 | "for problem, solution in solutions.items():\n", 422 | " print(f\"{problem}: {solution}\")\n", 423 | "print(\"=\"*60)" 424 | ] 425 | }, 426 | { 427 | "cell_type": "markdown", 428 | "metadata": {}, 429 | "source": [ 430 | "## Visualization: Complex Numbers on the Complex Plane" 431 | ] 432 | }, 433 | { 434 | "cell_type": "code", 435 | "execution_count": null, 436 | "metadata": {}, 437 | "outputs": [], 438 | "source": [ 439 | "# Visualize some of the results\n", 440 | "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))\n", 441 | "\n", 442 | "# Plot 1: Basic operations (Problems 1-8)\n", 443 | "results_basic = [z1, z2, z3, z4, z5, z6, z7, z8]\n", 444 | "labels_basic = ['P1', 'P2', 'P3', 'P4', 'P5', 'P6', 'P7', 'P8']\n", 445 | "\n", 446 | "for z, label in zip(results_basic, labels_basic):\n", 447 | " ax1.plot([0, z.real], [0, z.imag], 'o-', alpha=0.7, label=label)\n", 448 | " ax1.annotate(label, xy=(z.real, z.imag), xytext=(5, 5), \n", 449 | " textcoords='offset points', fontsize=9)\n", 450 | "\n", 451 | "ax1.axhline(y=0, color='k', linewidth=0.5)\n", 452 | "ax1.axvline(x=0, color='k', linewidth=0.5)\n", 453 | "ax1.grid(True, alpha=0.3)\n", 454 | "ax1.set_xlabel('Real Part')\n", 455 | "ax1.set_ylabel('Imaginary Part')\n", 456 | "ax1.set_title('Problems 1-8: Basic Operations')\n", 457 | "ax1.legend(loc='best', fontsize=8)\n", 458 | "ax1.set_aspect('equal')\n", 459 | "\n", 460 | "# Plot 2: Polar form examples (Problems 13-15)\n", 461 | "z13 = z_p13\n", 462 | "z14 = z_p14\n", 463 | "z15 = z_p15\n", 464 | "\n", 465 | "for z, label, color in [(z13, 'P13: 2+2√3i', 'blue'),\n", 466 | " (z14, 'P14: -5+5i', 'red'),\n", 467 | " (z15, 'P15: 3e^(iπ/2)', 'green')]:\n", 468 | " ax2.plot([0, z.real], [0, z.imag], 'o-', color=color, linewidth=2, \n", 469 | " markersize=8, label=label, alpha=0.7)\n", 470 | " \n", 471 | " # Draw the angle arc\n", 472 | " r = np.abs(z)\n", 473 | " theta = np.angle(z)\n", 474 | " arc_theta = np.linspace(0, theta, 50)\n", 475 | " arc_r = r * 0.3\n", 476 | " ax2.plot(arc_r * np.cos(arc_theta), arc_r * np.sin(arc_theta), \n", 477 | " '--', color=color, alpha=0.5)\n", 478 | "\n", 479 | "ax2.axhline(y=0, color='k', linewidth=0.5)\n", 480 | "ax2.axvline(x=0, color='k', linewidth=0.5)\n", 481 | "ax2.grid(True, alpha=0.3)\n", 482 | "ax2.set_xlabel('Real Part')\n", 483 | "ax2.set_ylabel('Imaginary Part')\n", 484 | "ax2.set_title('Problems 13-15: Polar Form')\n", 485 | "ax2.legend(loc='best')\n", 486 | "ax2.set_aspect('equal')\n", 487 | "\n", 488 | "plt.tight_layout()\n", 489 | "plt.show()" 490 | ] 491 | } 492 | ], 493 | "metadata": { 494 | "kernelspec": { 495 | "display_name": "Python 3", 496 | "language": "python", 497 | "name": "python3" 498 | }, 499 | "language_info": { 500 | "codemirror_mode": { 501 | "name": "ipython", 502 | "version": 3 503 | }, 504 | "file_extension": ".py", 505 | "mimetype": "text/x-python", 506 | "name": "python", 507 | "nbconvert_exporter": "python", 508 | "pygments_lexer": "ipython3", 509 | "version": "3.8.0" 510 | } 511 | }, 512 | "nbformat": 4, 513 | "nbformat_minor": 4 514 | } 515 | -------------------------------------------------------------------------------- /ComplexConjugate.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "id": "66e5c2ad", 6 | "metadata": {}, 7 | "source": [ 8 | "# COMPLEX CONJUGATE" 9 | ] 10 | }, 11 | { 12 | "cell_type": "markdown", 13 | "id": "b9a8be1b", 14 | "metadata": {}, 15 | "source": [ 16 | "## Package imports" 17 | ] 18 | }, 19 | { 20 | "cell_type": "code", 21 | "execution_count": 4, 22 | "id": "9b2c22f0", 23 | "metadata": {}, 24 | "outputs": [], 25 | "source": [ 26 | "from sympy import init_printing, symbols, I, re, im, conjugate, simplify" 27 | ] 28 | }, 29 | { 30 | "cell_type": "code", 31 | "execution_count": 5, 32 | "id": "53d10ad3", 33 | "metadata": {}, 34 | "outputs": [], 35 | "source": [ 36 | "init_printing()" 37 | ] 38 | }, 39 | { 40 | "cell_type": "markdown", 41 | "id": "3d6e1656", 42 | "metadata": {}, 43 | "source": [ 44 | "## Mathematical variables" 45 | ] 46 | }, 47 | { 48 | "cell_type": "code", 49 | "execution_count": 6, 50 | "id": "b58eed66", 51 | "metadata": {}, 52 | "outputs": [], 53 | "source": [ 54 | "x, y = symbols('x y', real=True)\n", 55 | "x1, x2, y1, y2 = symbols('x1 x2 y1 y2', real=True)\n" 56 | ] 57 | }, 58 | { 59 | "cell_type": "markdown", 60 | "id": "b46d70e6", 61 | "metadata": {}, 62 | "source": [ 63 | "## Definition" 64 | ] 65 | }, 66 | { 67 | "cell_type": "markdown", 68 | "id": "ff47b430", 69 | "metadata": {}, 70 | "source": [ 71 | "If $z = x+iy$ is a complex number, where $x$ and $y$ are real numbers and $i$ is the imaginary unit, then the **complex conjugate** of $z$, denoted by $\\overline{z}$, is defined as follows.\n", 72 | "$$\n", 73 | "\\overline{z} = x + i \\left( -y \\right) = x - iy\n", 74 | "$$" 75 | ] 76 | }, 77 | { 78 | "cell_type": "markdown", 79 | "id": "7d303860", 80 | "metadata": {}, 81 | "source": [ 82 | "## Properties" 83 | ] 84 | }, 85 | { 86 | "cell_type": "markdown", 87 | "id": "7e6ee619", 88 | "metadata": {}, 89 | "source": [ 90 | "### Addition" 91 | ] 92 | }, 93 | { 94 | "cell_type": "markdown", 95 | "id": "17d38631", 96 | "metadata": {}, 97 | "source": [ 98 | "$$\n", 99 | "\\overline{z_1 + z_2} = \\overline{z_1} + \\overline{z_2}\n", 100 | "$$" 101 | ] 102 | }, 103 | { 104 | "cell_type": "markdown", 105 | "id": "e4563cfd", 106 | "metadata": {}, 107 | "source": [ 108 | "__Proof__" 109 | ] 110 | }, 111 | { 112 | "cell_type": "markdown", 113 | "id": "0d247136", 114 | "metadata": {}, 115 | "source": [ 116 | "Let $z_{1} = x_{1} + i y_{1}$ and $z_{2} = x_{2} + i y_{2}$ be two complex numbers, where $x_{1}, x_{2}, y_{1}, y_{2} \\in \\mathbb{R}$. Then, by the definition of addition on the complex numbers, $z_{1} + z_{2} = \\left( x_{1} + x_{2} \\right) + i \\left( y_{1} + y_{2} \\right)$ and $\\overline{z_{1} + z_{2}} = \\left( x_{1} + x_{2} \\right) - i \\left( y_{1} + y_{2} \\right)$. On the other hand, by the definition of complex conjugate, we have that $\\overline{z_{1}} = x_{1} - i y_{1}$ and $\\overline{z_{2}} = x_{2} - i y_{2}$. Therefore, by the definition of addition on the complex numbers, $\\overline{z_{1}} + \\overline{z_{2}} = \\left( x_{1} + x_{2} \\right) - i \\left( y_{1} + y_{2} \\right)$. Hence, we conclude that $\\overline{z_{1} + z_{2}} = \\overline{z_{1}} + \\overline{z_{2}}$. $\\blacksquare$" 117 | ] 118 | }, 119 | { 120 | "cell_type": "code", 121 | "execution_count": 9, 122 | "id": "ab08b06c", 123 | "metadata": {}, 124 | "outputs": [], 125 | "source": [ 126 | "z1 = x1 + I*y1\n", 127 | "z2 = x2 + I*y2\n", 128 | "lhs = conjugate(z1 + z2)\n", 129 | "rhs = conjugate(z1) + conjugate(z2)" 130 | ] 131 | }, 132 | { 133 | "cell_type": "code", 134 | "execution_count": 11, 135 | "id": "d8b25ba0", 136 | "metadata": {}, 137 | "outputs": [ 138 | { 139 | "data": { 140 | "image/png": "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", 141 | "text/latex": [ 142 | "$\\displaystyle \\left( x_{1} + x_{2} - i y_{1} - i y_{2}, \\ x_{1} + x_{2} - i y_{1} - i y_{2}\\right)$" 143 | ], 144 | "text/plain": [ 145 | "(x₁ + x₂ - ⅈ⋅y₁ - ⅈ⋅y₂, x₁ + x₂ - ⅈ⋅y₁ - ⅈ⋅y₂)" 146 | ] 147 | }, 148 | "execution_count": 11, 149 | "metadata": {}, 150 | "output_type": "execute_result" 151 | } 152 | ], 153 | "source": [ 154 | "lhs.simplify(), rhs.simplify()" 155 | ] 156 | }, 157 | { 158 | "cell_type": "markdown", 159 | "id": "b1250a19", 160 | "metadata": {}, 161 | "source": [ 162 | "### Multiplication" 163 | ] 164 | }, 165 | { 166 | "cell_type": "markdown", 167 | "id": "50203189", 168 | "metadata": {}, 169 | "source": [ 170 | "$$\n", 171 | "\\overline{z_1 z_2} = \\overline{z_1} \\cdot \\overline{z_2}\n", 172 | "$$" 173 | ] 174 | }, 175 | { 176 | "cell_type": "markdown", 177 | "id": "82ae4d44", 178 | "metadata": {}, 179 | "source": [ 180 | "__Proof__" 181 | ] 182 | }, 183 | { 184 | "cell_type": "markdown", 185 | "id": "ea66778f", 186 | "metadata": {}, 187 | "source": [ 188 | "Let $z_{1} = x_{1} + i y_{1}$ and $z_{2} = x_{2} + i y_{2}$ be two complex numbers, where $x_{1}, x_{2}, y_{1}, y_{2} \\in \\mathbb{R}$. Then, by the definition of multiplication on the complex numbers, $z_{1} z_{2} = \\left( x_{1} x_{2} - y_{1} y_{2} \\right) + i \\left( x_{1} y_{2} + x_{2} y_{1} \\right)$ and $\\overline{z_{1} z_{2}} = \\left( x_{1} x_{2} - y_{1} y_{2} \\right) - i \\left( x_{1} y_{2} + x_{2} y_{1} \\right)$. On the other hand, by the definition of complex conjugate, we have that $\\overline{z_{1}} = x_{1} - i y_{1}$ and $\\overline{z_{2}} = x_{2} - i y_{2}$. Therefore, by the definition of multiplication on the complex numbers, $\\overline{z_{1}} \\cdot \\overline{z_{2}} = \\left( x_{1} x_{2} - y_{1} y_{2} \\right) - i \\left( x_{1} y_{2} + x_{2} y_{1} \\right)$. Hence, we conclude that $\\overline{z_{1} z_{2}} = \\overline{z_{1}} \\cdot \\overline{z_{2}}$. $\\blacksquare$" 189 | ] 190 | }, 191 | { 192 | "cell_type": "code", 193 | "execution_count": 12, 194 | "id": "a184e1e4", 195 | "metadata": {}, 196 | "outputs": [ 197 | { 198 | "data": { 199 | "image/png": "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", 200 | "text/latex": [ 201 | "$\\displaystyle \\left( \\left(x_{1} - i y_{1}\\right) \\left(x_{2} - i y_{2}\\right), \\ \\left(x_{1} - i y_{1}\\right) \\left(x_{2} - i y_{2}\\right)\\right)$" 202 | ], 203 | "text/plain": [ 204 | "((x₁ - ⅈ⋅y₁)⋅(x₂ - ⅈ⋅y₂), (x₁ - ⅈ⋅y₁)⋅(x₂ - ⅈ⋅y₂))" 205 | ] 206 | }, 207 | "execution_count": 12, 208 | "metadata": {}, 209 | "output_type": "execute_result" 210 | } 211 | ], 212 | "source": [ 213 | "lhs = conjugate(z1 * z2)\n", 214 | "rhs = conjugate(z1) * conjugate(z2)\n", 215 | "lhs.simplify(), rhs.simplify()" 216 | ] 217 | }, 218 | { 219 | "cell_type": "markdown", 220 | "id": "88f31ec9", 221 | "metadata": {}, 222 | "source": [ 223 | "### Re($z$)" 224 | ] 225 | }, 226 | { 227 | "cell_type": "markdown", 228 | "id": "4b168e31", 229 | "metadata": {}, 230 | "source": [ 231 | "$$\n", 232 | "\\text{Re}(z) = \\frac{z + \\overline{z}}{2}\n", 233 | "$$" 234 | ] 235 | }, 236 | { 237 | "cell_type": "markdown", 238 | "id": "52095a9b", 239 | "metadata": {}, 240 | "source": [ 241 | "__Proof__" 242 | ] 243 | }, 244 | { 245 | "cell_type": "markdown", 246 | "id": "0eba24ff", 247 | "metadata": {}, 248 | "source": [ 249 | "Let $z = x + i y$ be a complex number, where $x, y \\in \\mathbb{R}$. Then, by the definition of complex conjugate, we have that $\\overline{z} = x - i y$. Therefore, by the definition of addition on the complex numbers, we have that $z + \\overline{z} = \\left( x + x \\right) + i \\left( y - y \\right) = 2x + i \\cdot 0 = 2x$. Hence, by dividing both sides of the equation by $2$, we conclude that $\\frac{z + \\overline{z}}{2} = x$. Since, by the definition of real part, we have that $\\text{Re}(z) = x$, we conclude that $\\text{Re}(z) = \\frac{z + \\overline{z}}{2}$. $\\blacksquare$" 250 | ] 251 | }, 252 | { 253 | "cell_type": "code", 254 | "execution_count": 13, 255 | "id": "14ef365e", 256 | "metadata": {}, 257 | "outputs": [ 258 | { 259 | "data": { 260 | "image/png": "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", 261 | "text/latex": [ 262 | "$\\displaystyle x$" 263 | ], 264 | "text/plain": [ 265 | "x" 266 | ] 267 | }, 268 | "execution_count": 13, 269 | "metadata": {}, 270 | "output_type": "execute_result" 271 | } 272 | ], 273 | "source": [ 274 | "z = x + I*y\n", 275 | "re_z = re(z)\n", 276 | "re_z" 277 | ] 278 | }, 279 | { 280 | "cell_type": "code", 281 | "execution_count": 14, 282 | "id": "62a47687", 283 | "metadata": {}, 284 | "outputs": [ 285 | { 286 | "data": { 287 | "image/png": "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", 288 | "text/latex": [ 289 | "$\\displaystyle x$" 290 | ], 291 | "text/plain": [ 292 | "x" 293 | ] 294 | }, 295 | "execution_count": 14, 296 | "metadata": {}, 297 | "output_type": "execute_result" 298 | } 299 | ], 300 | "source": [ 301 | "(z + conjugate(z)) / 2" 302 | ] 303 | }, 304 | { 305 | "cell_type": "markdown", 306 | "id": "89b87102", 307 | "metadata": {}, 308 | "source": [ 309 | "### Im($z$)" 310 | ] 311 | }, 312 | { 313 | "cell_type": "markdown", 314 | "id": "ea930e08", 315 | "metadata": {}, 316 | "source": [ 317 | "$$\n", 318 | "\\text{Im}(z) = \\frac{z - \\overline{z}}{2i}\n", 319 | "$$" 320 | ] 321 | }, 322 | { 323 | "cell_type": "markdown", 324 | "id": "b25c2771", 325 | "metadata": {}, 326 | "source": [ 327 | "__Proof__" 328 | ] 329 | }, 330 | { 331 | "cell_type": "markdown", 332 | "id": "495efe4a", 333 | "metadata": {}, 334 | "source": [ 335 | "Let $z = x + iy$be a complex number, where $x, y$ are real numbers. Then, by the definition of subtraction on the complex numbers, we have that $z - \\overline{z} = \\left( x - x \\right) + i \\left( y - \\left( -y \\right) \\right) = 0 + i \\cdot 2y = i \\cdot 2y$. Hence, by dividing both sides of the equation by $2i$, we conclude that $\\frac{z - \\overline{z}}{2i} = y$. Since, by the definition of imaginary part, we have that $\\text{Im}(z) = y$, we conclude that $\\text{Im}(z) = \\frac{z - \\overline{z}}{2i}$. $\\blacksquare$" 336 | ] 337 | }, 338 | { 339 | "cell_type": "code", 340 | "execution_count": 15, 341 | "id": "06863a73", 342 | "metadata": {}, 343 | "outputs": [ 344 | { 345 | "data": { 346 | "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAAQCAYAAADNo/U5AAAAOnRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjEwLjcsIGh0dHBzOi8vbWF0cGxvdGxpYi5vcmcvTLEjVAAAAAlwSFlzAAASdAAAEnQB3mYfeAAAARRJREFUeJyV0r1KXFEUxfHfqJXmCcSIYGHeQF9ASGNpM4WdjVoY1EhA3OwqjdpoCBYWFj6A7+AHMpBiYGp7EdHGFIJjMTNyvdwR3c0+nLP+rLPOPrV2u+2zNdRbZOYadrAeEbtlYWZOoYmrgcL+WbfP9DHYxyBWitA//Md0hcs8ZvEnIpqvUEQ8oYGvmTlaAEawhxtsw4C3dV5xxW2MYTMiHt6Dprsu3/ADlzjuicrQBdoFp4Nu+OWIeJ1NrTynzGxhAos4wd+IWCpqyk50nn4Yh7jFVllQBfVyfcGviLj7CHTd7Q0cVZxXQht4VgrfF8rMOuZ0wjeqABjKzHHUMYkFtPCzH0Dnl3/Hb9zjFKsR8fge9AKLFU+fnbH1ggAAAABJRU5ErkJggg==", 347 | "text/latex": [ 348 | "$\\displaystyle y$" 349 | ], 350 | "text/plain": [ 351 | "y" 352 | ] 353 | }, 354 | "execution_count": 15, 355 | "metadata": {}, 356 | "output_type": "execute_result" 357 | } 358 | ], 359 | "source": [ 360 | "im_z = im(z)\n", 361 | "im_z" 362 | ] 363 | }, 364 | { 365 | "cell_type": "code", 366 | "execution_count": 16, 367 | "id": "99e92471", 368 | "metadata": {}, 369 | "outputs": [ 370 | { 371 | "data": { 372 | "image/png": "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", 373 | "text/latex": [ 374 | "$\\displaystyle y$" 375 | ], 376 | "text/plain": [ 377 | "y" 378 | ] 379 | }, 380 | "execution_count": 16, 381 | "metadata": {}, 382 | "output_type": "execute_result" 383 | } 384 | ], 385 | "source": [ 386 | "(z - conjugate(z)) / (2*I)" 387 | ] 388 | }, 389 | { 390 | "cell_type": "markdown", 391 | "id": "b9ff2b1d", 392 | "metadata": {}, 393 | "source": [] 394 | } 395 | ], 396 | "metadata": { 397 | "kernelspec": { 398 | "display_name": "mathematics_python3_13", 399 | "language": "python", 400 | "name": "python3" 401 | }, 402 | "language_info": { 403 | "codemirror_mode": { 404 | "name": "ipython", 405 | "version": 3 406 | }, 407 | "file_extension": ".py", 408 | "mimetype": "text/x-python", 409 | "name": "python", 410 | "nbconvert_exporter": "python", 411 | "pygments_lexer": "ipython3", 412 | "version": "3.13.9" 413 | } 414 | }, 415 | "nbformat": 4, 416 | "nbformat_minor": 5 417 | } 418 | -------------------------------------------------------------------------------- /VisualizingComplexNumbers.nb: -------------------------------------------------------------------------------- 1 | (* Content-type: application/vnd.wolfram.mathematica *) 2 | 3 | (*** Wolfram Notebook File ***) 4 | (* http://www.wolfram.com/nb *) 5 | 6 | (* CreatedBy='Wolfram 14.2' *) 7 | 8 | (*CacheID: 234*) 9 | (* Internal cache information: 10 | NotebookFileLineBreakTest 11 | NotebookFileLineBreakTest 12 | NotebookDataPosition[ 154, 7] 13 | NotebookDataLength[ 32970, 866] 14 | NotebookOptionsPosition[ 28775, 789] 15 | NotebookOutlinePosition[ 29206, 806] 16 | CellTagsIndexPosition[ 29163, 803] 17 | WindowFrame->Normal*) 18 | 19 | (* Beginning of Notebook Content *) 20 | Notebook[{ 21 | 22 | Cell[CellGroupData[{ 23 | Cell["VISUALIZING COMPLEX-VALUED FUNCTIONS", "Title", 24 | CellChangeTimes->{{3.948709959815542*^9, 3.948709972733034*^9}, { 25 | 3.94872815105126*^9, 26 | 3.9487281525324306`*^9}},ExpressionUUID->"1ce04e14-1e2f-47ba-b751-\ 27 | ca726ab4557a"], 28 | 29 | Cell["SINGLE COMPLEX VARIABLE FUNCTIONS IN THE ARGAND PLANE", "Subtitle", 30 | CellChangeTimes->{{3.948728206114085*^9, 31 | 3.948728228273488*^9}},ExpressionUUID->"bd5dba02-cb2b-4859-ae1e-\ 32 | a57dffc8f34f"], 33 | 34 | Cell[CellGroupData[{ 35 | 36 | Cell["Real-valued functions of a single variable", "Section", 37 | CellChangeTimes->{{3.948712300949502*^9, 3.9487123068766203`*^9}, { 38 | 3.9487133516819687`*^9, 39 | 3.9487133560884447`*^9}},ExpressionUUID->"f46c431e-aea3-4ea1-9b89-\ 40 | 1787f93cc4b7"], 41 | 42 | Cell[BoxData[ 43 | RowBox[{"y", "=", 44 | RowBox[{"f", 45 | RowBox[{"(", "x", ")"}]}]}]], "DisplayFormulaNumbered", 46 | CellChangeTimes->{ 47 | 3.9487279449940987`*^9},ExpressionUUID->"90ec4b92-5b3a-4f93-a7ab-\ 48 | e109392f4249"], 49 | 50 | Cell[BoxData[{ 51 | RowBox[{ 52 | RowBox[{ 53 | RowBox[{"f", ":", " ", "X"}], "->", "Y"}], ",", " ", 54 | RowBox[{"X", "=", "\[DoubleStruckCapitalR]"}], ",", 55 | RowBox[{"Y", "=", "\[DoubleStruckCapitalR]"}]}], "\[IndentingNewLine]", 56 | RowBox[{ 57 | RowBox[{"f", ":", " ", "\[DoubleStruckCapitalR]"}], "->", 58 | "\[DoubleStruckCapitalR]"}], "\[IndentingNewLine]", 59 | RowBox[{ 60 | RowBox[{"f", ":", " ", 61 | RowBox[{"x", "\[RightTeeArrow]", "y"}]}], ",", " ", 62 | RowBox[{"x", "\[Element]", "X"}], ",", 63 | RowBox[{"y", "\[Element]", "Y"}]}]}], "DisplayFormulaNumbered", 64 | CellChangeTimes->{{3.9487267296542873`*^9, 3.94872682547956*^9}, { 65 | 3.9487279093575163`*^9, 66 | 3.948727938199629*^9}},ExpressionUUID->"c78f3e59-1bd8-43a6-8451-\ 67 | 771898c4ede6"], 68 | 69 | Cell[CellGroupData[{ 70 | 71 | Cell["Cartesian plane", "Subsection", 72 | CellChangeTimes->{{3.948727812688085*^9, 73 | 3.9487278165849543`*^9}},ExpressionUUID->"a889d8bb-5e29-41d0-9886-\ 74 | b8ce9d753b0c"], 75 | 76 | Cell[TextData[{ 77 | "We are familiar with the visualization of the single-variable function ", 78 | Cell[BoxData[ 79 | FormBox[ 80 | TemplateBox[<|"boxes" -> FormBox[ 81 | RowBox[{ 82 | StyleBox["f", "TI"], 83 | RowBox[{"(", 84 | StyleBox["x", "TI"], ")"}], "\[LongEqual]", 85 | SuperscriptBox[ 86 | StyleBox["x", "TI"], "3"]}], TraditionalForm], "errors" -> {}, 87 | "input" -> "f(x)=x^{3}", "state" -> "Boxes"|>, 88 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 89 | "96b7232f-5b96-4278-951d-b7554fdaf370"], 90 | "." 91 | }], "Text", 92 | CellChangeTimes->{{3.9487126407879143`*^9, 93 | 3.94871267790917*^9}},ExpressionUUID->"6ee17c6b-8629-43ba-81d3-\ 94 | 8787b93ecbb6"], 95 | 96 | Cell[BoxData[ 97 | RowBox[{ 98 | RowBox[{"y", "=", 99 | RowBox[{ 100 | RowBox[{"f", 101 | RowBox[{"(", "x", ")"}]}], "=", 102 | SuperscriptBox["x", "3"]}]}], ",", " ", 103 | RowBox[{"x", "\[Element]", "\[DoubleStruckCapitalR]"}], ",", " ", 104 | RowBox[{ 105 | "y", "\[Element]", "\[DoubleStruckCapitalR]"}]}]], "DisplayFormulaNumbered",\ 106 | 107 | CellChangeTimes->{{3.948726853015891*^9, 3.948726855413084*^9}, { 108 | 3.948727889029504*^9, 109 | 3.948727905983781*^9}},ExpressionUUID->"2e2bde69-21f8-4eb6-b31d-\ 110 | fa7a1fd6971e"], 111 | 112 | Cell[BoxData[ 113 | RowBox[{ 114 | RowBox[{"f", "[", "x_", "]"}], ":=", 115 | SuperscriptBox["x", "3"]}]], "Input", 116 | CellChangeTimes->{{3.9487132477413483`*^9, 3.9487132525780363`*^9}}, 117 | CellLabel->"In[19]:=",ExpressionUUID->"734b99f0-3e81-4efd-acc8-fc13120eb372"], 118 | 119 | Cell[BoxData[ 120 | RowBox[{"Manipulate", "[", 121 | RowBox[{ 122 | RowBox[{"Plot", "[", 123 | RowBox[{ 124 | RowBox[{"f", "[", "x", "]"}], ",", 125 | RowBox[{"{", 126 | RowBox[{"x", ",", 127 | RowBox[{"-", "3"}], ",", "t"}], "}"}], ",", 128 | RowBox[{"PlotRange", "->", 129 | RowBox[{"{", 130 | RowBox[{ 131 | RowBox[{"{", 132 | RowBox[{ 133 | RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", 134 | RowBox[{"{", 135 | RowBox[{ 136 | RowBox[{"-", "30"}], ",", "30"}], "}"}]}], "}"}]}], ",", 137 | RowBox[{"ImageSize", "->", "Large"}], ",", 138 | RowBox[{"GridLines", "->", "Automatic"}], ",", 139 | RowBox[{"AxesLabel", "->", 140 | RowBox[{"{", 141 | RowBox[{"\"\<\!\(\*Cell[\"x\[Element]X\",ExpressionUUID->\"98db33a6-\ 142 | a28a-431f-8ec8-f0d2f4df9516\"]\)\>\"", 143 | ",", "\"\<\!\(\*Cell[\"y\[Element]Y\",ExpressionUUID->\"17046015-b42d-\ 144 | 4c7e-8e09-ddfed46dcdce\"]\)\>\""}], "}"}]}]}], "]"}], ",", 145 | RowBox[{"{", 146 | RowBox[{"t", ",", 147 | RowBox[{"-", "2.9"}], ",", "3"}], "}"}]}], "]"}]], "Input", 148 | CellChangeTimes->{{3.948712726861272*^9, 3.948712817491972*^9}, { 149 | 3.948713258945887*^9, 3.948713261275279*^9}, {3.948727459651786*^9, 150 | 3.948727529024826*^9}, {3.948728658011351*^9, 3.948728659547921*^9}}, 151 | CellLabel->"In[20]:=",ExpressionUUID->"10c8e7d4-0ddc-4f0f-a1f8-9fc4ea439f10"] 152 | }, Closed]], 153 | 154 | Cell[CellGroupData[{ 155 | 156 | Cell["Mapping", "Subsection", 157 | CellChangeTimes->{{3.948727825203271*^9, 158 | 3.948727834157699*^9}},ExpressionUUID->"d2c149d2-e2e8-42d7-a1f5-\ 159 | eb5919a738cb"], 160 | 161 | Cell[BoxData[ 162 | RowBox[{"Manipulate", "[", 163 | RowBox[{ 164 | RowBox[{"Show", "[", 165 | RowBox[{ 166 | RowBox[{"ListPlot", "[", 167 | RowBox[{ 168 | RowBox[{"{", 169 | RowBox[{"{", 170 | RowBox[{"x", ",", "0"}], "}"}], "}"}], ",", 171 | RowBox[{"PlotLegends", "->", 172 | RowBox[{ 173 | "{", "\"\<\!\(\*Cell[\"x\[Element]X\",ExpressionUUID->\"ad4c1e3c-fd7e-\ 174 | 47f8-8bd7-255df74deed8\"]\)\>\"", "}"}]}], ",", 175 | RowBox[{"GridLines", "->", 176 | RowBox[{"{", 177 | RowBox[{ 178 | RowBox[{"Range", "[", 179 | RowBox[{ 180 | RowBox[{"-", "8"}], ",", "8", ",", "0.5"}], "]"}], ",", 181 | RowBox[{"{", "}"}]}], "}"}]}], ",", 182 | RowBox[{"Ticks", "->", 183 | RowBox[{"{", 184 | RowBox[{ 185 | RowBox[{"Range", "[", 186 | RowBox[{ 187 | RowBox[{"-", "8"}], ",", "8", ",", "1"}], "]"}], ",", "None"}], 188 | "}"}]}], ",", 189 | RowBox[{"ImageSize", "->", "Large"}], ",", 190 | RowBox[{"PlotRange", "->", 191 | RowBox[{"{", 192 | RowBox[{ 193 | RowBox[{"{", 194 | RowBox[{ 195 | RowBox[{"-", "8.4"}], ",", "8.4"}], "}"}], ",", 196 | RowBox[{"{", 197 | RowBox[{ 198 | RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}], ",", 199 | RowBox[{"PlotMarkers", "->", 200 | RowBox[{"{", 201 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}]}], "]"}], ",", 202 | RowBox[{"ListPlot", "[", 203 | RowBox[{ 204 | RowBox[{"{", 205 | RowBox[{"{", 206 | RowBox[{ 207 | RowBox[{"f", "[", "x", "]"}], ",", "0"}], "}"}], "}"}], ",", 208 | RowBox[{"PlotLegends", "->", 209 | RowBox[{ 210 | "{", "\"\<\!\(\*Cell[\"y\[Element]Y\",ExpressionUUID->\"cdb42e2d-13cb-\ 211 | 46c3-9923-de3385dd1842\"]\)\>\"", "}"}]}], ",", 212 | RowBox[{"PlotMarkers", "->", 213 | RowBox[{"{", 214 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 215 | RowBox[{"PlotStyle", "->", "Orange"}]}], "]"}]}], "]"}], ",", 216 | RowBox[{"{", 217 | RowBox[{ 218 | RowBox[{"{", 219 | RowBox[{"x", ",", "0"}], "}"}], ",", 220 | RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]], "Input", 221 | CellChangeTimes->{{3.948727028837247*^9, 3.948727246140099*^9}, { 222 | 3.9487273031653223`*^9, 3.948727393711205*^9}, {3.9487274257380533`*^9, 223 | 3.9487274308167562`*^9}, {3.948727571258555*^9, 3.948727579746594*^9}, { 224 | 3.948727667047694*^9, 3.948727740145804*^9}, {3.948727775430665*^9, 225 | 3.948727777719912*^9}, {3.948728678714323*^9, 3.948728684525996*^9}}, 226 | CellLabel->"In[21]:=",ExpressionUUID->"55811f37-4b56-4dbc-8d9c-b26be6f45143"], 227 | 228 | Cell[BoxData[ 229 | RowBox[{"Manipulate", "[", 230 | RowBox[{ 231 | RowBox[{"Column", "[", 232 | RowBox[{ 233 | RowBox[{"{", 234 | RowBox[{ 235 | RowBox[{"ListPlot", "[", 236 | RowBox[{ 237 | RowBox[{"{", 238 | RowBox[{"{", 239 | RowBox[{"x", ",", "0"}], "}"}], "}"}], ",", 240 | RowBox[{"Ticks", "->", 241 | RowBox[{"{", 242 | RowBox[{ 243 | RowBox[{"Range", "[", 244 | RowBox[{ 245 | RowBox[{"-", "2"}], ",", "2", ",", 246 | FractionBox["1", "2"]}], "]"}], ",", "None"}], "}"}]}], ",", 247 | RowBox[{"PlotMarkers", "\[Rule]", 248 | RowBox[{"{", 249 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 250 | RowBox[{ 251 | "PlotLabel", 252 | "->", "\"\<\!\(\*Cell[\"x\[Element]X\",ExpressionUUID->\"a9e95e00-\ 253 | 978e-40dd-9e3f-c85b7a621e57\"]\)\>\""}], ",", 254 | RowBox[{"GridLines", "->", 255 | RowBox[{"{", 256 | RowBox[{ 257 | RowBox[{"Range", "[", 258 | RowBox[{ 259 | RowBox[{"-", "2"}], ",", "2", ",", "0.2"}], "]"}], ",", 260 | RowBox[{"{", "}"}]}], "}"}]}], ",", 261 | RowBox[{"ImageSize", "->", "Large"}], ",", 262 | RowBox[{"PlotRange", "->", 263 | RowBox[{"{", 264 | RowBox[{ 265 | RowBox[{"{", 266 | RowBox[{ 267 | RowBox[{"-", "2.1"}], ",", "2.1"}], "}"}], ",", 268 | RowBox[{"{", 269 | RowBox[{ 270 | RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}]}], "]"}], ",", 271 | 272 | RowBox[{"ListPlot", "[", 273 | RowBox[{ 274 | RowBox[{"{", 275 | RowBox[{"{", 276 | RowBox[{ 277 | RowBox[{"f", "[", "x", "]"}], ",", "0"}], "}"}], "}"}], ",", 278 | RowBox[{"Ticks", "->", 279 | RowBox[{"{", 280 | RowBox[{ 281 | RowBox[{"Range", "[", 282 | RowBox[{ 283 | RowBox[{"-", "8"}], ",", "8", ",", "1"}], "]"}], ",", "None"}], 284 | "}"}]}], ",", 285 | RowBox[{"PlotMarkers", "\[Rule]", 286 | RowBox[{"{", 287 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 288 | RowBox[{ 289 | "PlotLabel", 290 | "->", "\"\<\!\(\*Cell[TextData[Cell[BoxData[FormBox[RowBox[{\"y\", \ 291 | \"\[Element]\", \"Y\"}], \ 292 | TraditionalForm]],ExpressionUUID->\"8f70938f-6b9e-4c47-b2a6-0226eed933f7\"]],\ 293 | ExpressionUUID->\"663f9d0d-0c01-4f4a-ba5e-535f6fdb1c3b\"]\)\>\""}], ",", 294 | RowBox[{"GridLines", "->", 295 | RowBox[{"{", 296 | RowBox[{ 297 | RowBox[{"Range", "[", 298 | RowBox[{ 299 | RowBox[{"-", "8"}], ",", "8", ",", "0.2"}], "]"}], ",", 300 | RowBox[{"{", "}"}]}], "}"}]}], ",", 301 | RowBox[{"ImageSize", "->", "Large"}], ",", 302 | RowBox[{"PlotRange", "->", 303 | RowBox[{"{", 304 | RowBox[{ 305 | RowBox[{"{", 306 | RowBox[{ 307 | RowBox[{"-", "8.3"}], ",", "8.3"}], "}"}], ",", 308 | RowBox[{"{", 309 | RowBox[{ 310 | RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "}"}]}]}], "]"}]}], 311 | "}"}], ",", "\"\< \>\""}], "]"}], ",", 312 | RowBox[{"{", 313 | RowBox[{ 314 | RowBox[{"{", 315 | RowBox[{"x", ",", "0"}], "}"}], ",", 316 | RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]], "Input", 317 | CellChangeTimes->{{3.9487090032059097`*^9, 3.948709095674168*^9}, { 318 | 3.948709159601325*^9, 3.948709169553012*^9}, {3.948710136045371*^9, 319 | 3.948710145784525*^9}, {3.9487103403354177`*^9, 3.9487104190023527`*^9}, { 320 | 3.948710457678914*^9, 3.9487104708278008`*^9}, {3.948710727733835*^9, 321 | 3.948710736161688*^9}, 3.9487113042576027`*^9, {3.948712843879509*^9, 322 | 3.9487129947133083`*^9}, {3.948713059249504*^9, 3.9487130622366543`*^9}, { 323 | 3.94871310413085*^9, 3.9487131502033243`*^9}, {3.9487131921798162`*^9, 324 | 3.948713205154139*^9}, {3.948713267152307*^9, 3.948713270223675*^9}, { 325 | 3.948713308498906*^9, 3.948713330480584*^9}, {3.948726419684766*^9, 326 | 3.948726465121935*^9}, {3.9487265625248547`*^9, 3.948726568136199*^9}, { 327 | 3.948726622303492*^9, 3.9487266738535748`*^9}, {3.948726934830558*^9, 328 | 3.948726983248193*^9}, {3.948728044666383*^9, 3.948728065078863*^9}, { 329 | 3.9487287134621677`*^9, 3.948728717239811*^9}, {3.948728773310528*^9, 330 | 3.948728774436758*^9}, {3.9488010703177757`*^9, 3.948801072035036*^9}}, 331 | CellLabel->"In[23]:=",ExpressionUUID->"45018dcf-643d-447a-9cab-b9a1114abf68"] 332 | }, Closed]] 333 | }, Closed]], 334 | 335 | Cell[CellGroupData[{ 336 | 337 | Cell["Complex conjugate", "Section", 338 | CellChangeTimes->{{3.9487099885578947`*^9, 339 | 3.948709991563753*^9}},ExpressionUUID->"d1e63113-27ff-4d5b-8afa-\ 340 | ac7357c6cc11"], 341 | 342 | Cell[BoxData[ 343 | RowBox[{ 344 | RowBox[{ 345 | RowBox[{"f", 346 | RowBox[{"(", "z", ")"}]}], "=", 347 | OverscriptBox["z", "_"]}], ",", " ", 348 | RowBox[{"z", "\[Element]", "\[DoubleStruckCapitalC]"}], ",", " ", 349 | RowBox[{"z", "=", 350 | RowBox[{"x", "+", "iy"}]}], ",", " ", "x", ",", 351 | RowBox[{ 352 | "y", "\[Element]", "\[DoubleStruckCapitalR]"}]}]], "DisplayFormulaNumbered",\ 353 | 354 | CellChangeTimes->{{3.948710013208673*^9, 355 | 3.9487100537665586`*^9}},ExpressionUUID->"3ade09d5-be72-42ba-9193-\ 356 | d054e23e04fa"], 357 | 358 | Cell[BoxData[ 359 | RowBox[{ 360 | RowBox[{"f", "[", 361 | RowBox[{"x_", ",", "y_"}], "]"}], ":=", 362 | RowBox[{"Conjugate", "[", 363 | RowBox[{"x", "+", 364 | RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}]}]], "Input", 365 | CellChangeTimes->{{3.9487089496950607`*^9, 3.948708979904636*^9}, { 366 | 3.948711529599378*^9, 3.9487115352200527`*^9}}, 367 | CellLabel->"In[24]:=",ExpressionUUID->"b908dc19-4a75-4857-aa52-11fbe8bfe12a"], 368 | 369 | Cell[CellGroupData[{ 370 | 371 | Cell["Visualizing a specific complex number", "Subsection", 372 | CellChangeTimes->{{3.9487110755560637`*^9, 373 | 3.9487111024375887`*^9}},ExpressionUUID->"b229a6d3-126f-44ed-a825-\ 374 | 0d99b0a03b32"], 375 | 376 | Cell[BoxData[ 377 | RowBox[{"Manipulate", "[", 378 | RowBox[{ 379 | RowBox[{"Row", "[", 380 | RowBox[{ 381 | RowBox[{"{", 382 | RowBox[{ 383 | RowBox[{"ListPlot", "[", 384 | RowBox[{ 385 | RowBox[{"{", 386 | RowBox[{"{", 387 | RowBox[{"x", ",", "y"}], "}"}], "}"}], ",", 388 | RowBox[{"PlotMarkers", "\[Rule]", 389 | RowBox[{"{", 390 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 391 | RowBox[{ 392 | "PlotLabel", 393 | "->", "\"\<\!\(\*Cell[\"z=x+iy\",ExpressionUUID->\"48b9dc5d-7fce-\ 394 | 4718-b55f-f9767bb3096c\"]\)\>\""}], ",", 395 | RowBox[{"GridLines", "->", "Automatic"}], ",", 396 | RowBox[{"ImageSize", "->", "Large"}], ",", 397 | RowBox[{"PlotRange", "->", 398 | RowBox[{"{", 399 | RowBox[{ 400 | RowBox[{"{", 401 | RowBox[{ 402 | RowBox[{"-", "2.1"}], ",", "2.1"}], "}"}], ",", 403 | RowBox[{"{", 404 | RowBox[{ 405 | RowBox[{"-", "2.1"}], ",", "2.1"}], "}"}]}], "}"}]}]}], "]"}], ",", 406 | RowBox[{"ListPlot", "[", 407 | RowBox[{ 408 | RowBox[{"{", 409 | RowBox[{"{", 410 | RowBox[{ 411 | RowBox[{"Re", "[", 412 | RowBox[{"f", "[", 413 | RowBox[{"x", ",", "y"}], "]"}], "]"}], ",", 414 | RowBox[{"Im", "[", 415 | RowBox[{"f", "[", 416 | RowBox[{"x", ",", "y"}], "]"}], "]"}]}], "}"}], "}"}], ",", 417 | RowBox[{"PlotMarkers", "\[Rule]", 418 | RowBox[{"{", 419 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 420 | RowBox[{"PlotLabel", "->", "\"\<\!\(\*Cell[TextData[{ 421 | \"f(z)=\", 422 | Cell[BoxData[FormBox[ 423 | OverscriptBox[\"z\", \"_\"], \ 424 | TraditionalForm]],ExpressionUUID->\"351d704d-041f-4731-8e2b-18825d68908f\"] 425 | }],ExpressionUUID->\"852d78f7-4f19-40f6-a92e-547c467a9607\"]\)\>\""}], ",", 426 | RowBox[{"GridLines", "->", "Automatic"}], ",", 427 | RowBox[{"ImageSize", "->", "Large"}], ",", 428 | RowBox[{"PlotRange", "->", 429 | RowBox[{"{", 430 | RowBox[{ 431 | RowBox[{"{", 432 | RowBox[{ 433 | RowBox[{"-", "2.1"}], ",", "2.1"}], "}"}], ",", 434 | RowBox[{"{", 435 | RowBox[{ 436 | RowBox[{"-", "2.1"}], ",", "2.1"}], "}"}]}], "}"}]}]}], "]"}]}], 437 | "}"}], ",", "\"\< \>\""}], "]"}], ",", 438 | RowBox[{"{", 439 | RowBox[{ 440 | RowBox[{"{", 441 | RowBox[{"x", ",", "0"}], "}"}], ",", 442 | RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 443 | RowBox[{"{", 444 | RowBox[{ 445 | RowBox[{"{", 446 | RowBox[{"y", ",", "0"}], "}"}], ",", 447 | RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]], "Input", 448 | CellChangeTimes->{{3.9487090032059097`*^9, 3.948709095674168*^9}, { 449 | 3.948709159601325*^9, 3.948709169553012*^9}, {3.948710136045371*^9, 450 | 3.948710145784525*^9}, {3.9487103403354177`*^9, 3.9487104190023527`*^9}, { 451 | 3.948710457678914*^9, 3.9487104708278008`*^9}, {3.948710727733835*^9, 452 | 3.948710736161688*^9}, 3.9487113042576027`*^9, {3.9487287947807083`*^9, 453 | 3.948728801344672*^9}, {3.948728857727688*^9, 3.948728859065795*^9}}, 454 | CellLabel->"In[25]:=",ExpressionUUID->"ab63d960-ee5c-4f90-85a6-b4420580a7bd"] 455 | }, Closed]], 456 | 457 | Cell[CellGroupData[{ 458 | 459 | Cell["Stream plot of complex numbers", "Subsection", 460 | CellChangeTimes->{{3.94871110685332*^9, 461 | 3.9487111158408422`*^9}},ExpressionUUID->"81577657-12e2-49bb-8fb5-\ 462 | 9e51f3537e57"], 463 | 464 | Cell[BoxData[ 465 | RowBox[{"StreamPlot", "[", 466 | RowBox[{ 467 | RowBox[{"{", 468 | RowBox[{"0", ",", 469 | RowBox[{"Im", "[", 470 | RowBox[{"Conjugate", "[", 471 | RowBox[{"x", "+", 472 | RowBox[{"\[ImaginaryI]", " ", "y"}]}], "]"}], "]"}]}], "}"}], ",", 473 | RowBox[{"{", 474 | RowBox[{"x", ",", 475 | RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 476 | RowBox[{"{", 477 | RowBox[{"y", ",", 478 | RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 479 | RowBox[{ 480 | "PlotLabel", 481 | "->", "\"\<\!\(\*Cell[TextData[Cell[BoxData[FormBox[RowBox[{RowBox[{\"f\",\ 482 | \"(\", \"z\", \")\"}], \"=\", 483 | OverscriptBox[\"z\", \"_\"]}], \ 484 | TraditionalForm]],ExpressionUUID->\"942b41f1-c449-46c0-acf8-5488e2b9b636\"]],\ 485 | ExpressionUUID->\"f67343b2-3c84-465d-bf35-83f901bd514e\"]\)\>\""}], ",", 486 | RowBox[{"PerformanceGoal", "->", "\"\\""}], ",", 487 | RowBox[{"ImageSize", "->", "Large"}], ",", 488 | RowBox[{"PlotLegends", "->", "Automatic"}]}], "]"}]], "Input", 489 | CellChangeTimes->{{3.948707503167643*^9, 3.948707565465602*^9}, { 490 | 3.948707598165729*^9, 3.9487076614917994`*^9}, {3.94870792395779*^9, 491 | 3.948707968292075*^9}, {3.9487082065953817`*^9, 3.948708291937604*^9}, { 492 | 3.948709376587838*^9, 3.9487094266633167`*^9}, {3.9487094965748*^9, 493 | 3.948709512266919*^9}, {3.9487102191746492`*^9, 3.94871023443944*^9}}, 494 | CellLabel->"In[26]:=",ExpressionUUID->"0e80556d-da23-4b69-8e32-2cf6bc8921f9"] 495 | }, Closed]], 496 | 497 | Cell[CellGroupData[{ 498 | 499 | Cell["Complex plot", "Subsection", 500 | CellChangeTimes->{{3.948711128869843*^9, 501 | 3.948711133106535*^9}},ExpressionUUID->"a83c6484-a5af-4420-a3fb-\ 502 | 8045a0c48641"], 503 | 504 | Cell[BoxData[ 505 | RowBox[{"ComplexPlot", "[", 506 | RowBox[{ 507 | RowBox[{"Conjugate", "[", "z", "]"}], ",", 508 | RowBox[{"{", 509 | RowBox[{"z", ",", 510 | RowBox[{ 511 | RowBox[{"-", "2"}], "-", 512 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ",", 513 | RowBox[{"2", "+", 514 | RowBox[{"2", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", 515 | RowBox[{"PlotLegends", "->", 516 | RowBox[{"Placed", "[", 517 | RowBox[{"Automatic", ",", " ", "Left"}], "]"}]}], ",", 518 | RowBox[{"AxesLabel", "->", 519 | RowBox[{"{", 520 | RowBox[{"\"\\"", ",", "\"\\"", 521 | ",", "\"\\""}], "}"}]}], ",", 522 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]], "Input", 523 | CellChangeTimes->{{3.948711006131954*^9, 3.9487110062893972`*^9}, 524 | 3.948728951743556*^9}, 525 | CellLabel->"In[27]:=",ExpressionUUID->"aabb94a3-dfe3-493d-ad9a-f94c3275424e"] 526 | }, Closed]], 527 | 528 | Cell[CellGroupData[{ 529 | 530 | Cell["3D complex plot", "Subsection", 531 | CellChangeTimes->{{3.948711138755513*^9, 532 | 3.948711141721005*^9}},ExpressionUUID->"21098d3d-267e-4fe4-8075-\ 533 | 8c7c9616ca7f"], 534 | 535 | Cell[BoxData[ 536 | RowBox[{"ComplexPlot3D", "[", 537 | RowBox[{ 538 | RowBox[{"Conjugate", "[", "z", "]"}], ",", 539 | RowBox[{"{", 540 | RowBox[{"z", ",", 541 | RowBox[{ 542 | RowBox[{"-", "2"}], "-", 543 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ",", 544 | RowBox[{"2", "+", 545 | RowBox[{"2", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", 546 | RowBox[{"PlotLegends", "->", 547 | RowBox[{"Placed", "[", 548 | RowBox[{"Automatic", ",", " ", "Left"}], "]"}]}], ",", 549 | RowBox[{"AxesLabel", "->", 550 | RowBox[{"{", 551 | RowBox[{"\"\\"", ",", "\"\\"", 552 | ",", "\"\\""}], "}"}]}], ",", 553 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]], "Input", 554 | CellChangeTimes->{{3.948706707620935*^9, 3.9487067655963297`*^9}, { 555 | 3.948707056938891*^9, 3.948707073987926*^9}, {3.9487072051186438`*^9, 556 | 3.948707220088271*^9}, {3.948707255284111*^9, 3.948707286495392*^9}, { 557 | 3.948707431978191*^9, 3.948707480302042*^9}, {3.948709261516107*^9, 558 | 3.948709309365879*^9}, {3.948710278763673*^9, 3.948710282342723*^9}, { 559 | 3.948710494834928*^9, 3.948710505854477*^9}, 3.948728954760166*^9}, 560 | CellLabel->"In[28]:=",ExpressionUUID->"c6b14664-0064-4acd-bc9e-39cc5f5978eb"] 561 | }, Closed]] 562 | }, Closed]], 563 | 564 | Cell[CellGroupData[{ 565 | 566 | Cell["Square of the complex number", "Section", 567 | CellChangeTimes->{{3.948710558602105*^9, 568 | 3.9487105637300367`*^9}},ExpressionUUID->"9081f0aa-b848-483d-89ac-\ 569 | 732f3885a2d2"], 570 | 571 | Cell[BoxData[ 572 | RowBox[{ 573 | RowBox[{ 574 | RowBox[{"f", 575 | RowBox[{"(", "z", ")"}]}], "=", 576 | SuperscriptBox["z", "2"]}], ",", " ", 577 | RowBox[{"z", "\[Element]", "\[DoubleStruckCapitalC]"}], ",", " ", 578 | RowBox[{"z", "=", 579 | RowBox[{"x", "+", "iy"}]}], ",", " ", "x", ",", 580 | RowBox[{ 581 | "y", "\[Element]", "\[DoubleStruckCapitalR]"}]}]], "DisplayFormulaNumbered",\ 582 | 583 | CellChangeTimes->{{3.9487105760869513`*^9, 584 | 3.948710592560792*^9}},ExpressionUUID->"3edba2ff-067a-4592-8813-\ 585 | fdeb35bfd5fb"], 586 | 587 | Cell[BoxData[ 588 | RowBox[{ 589 | RowBox[{"f", "[", 590 | RowBox[{"x_", ",", "y_"}], "]"}], ":=", 591 | SuperscriptBox[ 592 | RowBox[{"(", 593 | RowBox[{"x", "+", 594 | RowBox[{"y", " ", "\[ImaginaryI]"}]}], ")"}], "2"]}]], "Input", 595 | CellChangeTimes->{{3.9487089496950607`*^9, 3.948708979904636*^9}, { 596 | 3.948711461875989*^9, 3.948711468334682*^9}}, 597 | CellLabel->"In[29]:=",ExpressionUUID->"705697d6-2b7c-423a-b71c-8b9a2a6f28c5"], 598 | 599 | Cell[CellGroupData[{ 600 | 601 | Cell["Visualizing a specific complex number", "Subsection", 602 | CellChangeTimes->{{3.948711320458827*^9, 603 | 3.948711327527321*^9}},ExpressionUUID->"72768586-42e6-400d-a77e-\ 604 | e3813d7e895e"], 605 | 606 | Cell[BoxData[ 607 | RowBox[{"Manipulate", "[", 608 | RowBox[{ 609 | RowBox[{"Row", "[", 610 | RowBox[{ 611 | RowBox[{"{", 612 | RowBox[{ 613 | RowBox[{"ListPlot", "[", 614 | RowBox[{ 615 | RowBox[{"{", 616 | RowBox[{"{", 617 | RowBox[{"x", ",", "y"}], "}"}], "}"}], ",", 618 | RowBox[{"GridLines", "->", "Automatic"}], ",", 619 | RowBox[{"PlotMarkers", "\[Rule]", 620 | RowBox[{"{", 621 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 622 | RowBox[{"ImageSize", "->", "Large"}], ",", 623 | RowBox[{"PlotRange", "->", 624 | RowBox[{"{", 625 | RowBox[{ 626 | RowBox[{"{", 627 | RowBox[{ 628 | RowBox[{"-", "2.2"}], ",", "2.2"}], "}"}], ",", 629 | RowBox[{"{", 630 | RowBox[{ 631 | RowBox[{"-", "2.2"}], ",", "2.2"}], "}"}]}], "}"}]}]}], "]"}], ",", 632 | RowBox[{"ListPlot", "[", 633 | RowBox[{ 634 | RowBox[{"{", 635 | RowBox[{"{", 636 | RowBox[{ 637 | RowBox[{"Re", "[", 638 | RowBox[{"f", "[", 639 | RowBox[{"x", ",", "y"}], "]"}], "]"}], ",", 640 | RowBox[{"Im", "[", 641 | RowBox[{"f", "[", 642 | RowBox[{"x", ",", "y"}], "]"}], "]"}]}], "}"}], "}"}], ",", 643 | RowBox[{"GridLines", "->", "Automatic"}], ",", 644 | RowBox[{"PlotMarkers", "\[Rule]", 645 | RowBox[{"{", 646 | RowBox[{"\"\\"", ",", "Large"}], "}"}]}], ",", 647 | RowBox[{"ImageSize", "->", "Large"}], ",", 648 | RowBox[{"PlotRange", "->", 649 | RowBox[{"{", 650 | RowBox[{ 651 | RowBox[{"{", 652 | RowBox[{ 653 | RowBox[{"-", "5.2"}], ",", "5.2"}], "}"}], ",", 654 | RowBox[{"{", 655 | RowBox[{ 656 | RowBox[{"-", "5.2"}], ",", "5.2"}], "}"}]}], "}"}]}]}], "]"}]}], 657 | "}"}], ",", "\"\< \>\""}], "]"}], ",", 658 | RowBox[{"{", 659 | RowBox[{ 660 | RowBox[{"{", 661 | RowBox[{"x", ",", "0"}], "}"}], ",", 662 | RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 663 | RowBox[{"{", 664 | RowBox[{ 665 | RowBox[{"{", 666 | RowBox[{"y", ",", "0"}], "}"}], ",", 667 | RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]], "Input", 668 | CellChangeTimes->{{3.948708316892497*^9, 3.948708586348092*^9}, { 669 | 3.948708616896469*^9, 3.948708634763012*^9}, {3.948708681133431*^9, 670 | 3.948708738796996*^9}, {3.948709765514031*^9, 3.948709809190874*^9}, { 671 | 3.948709895327739*^9, 3.948709918918098*^9}, {3.948710787567266*^9, 672 | 3.9487108275495768`*^9}, {3.948711195288034*^9, 3.948711219178328*^9}, { 673 | 3.948711251989037*^9, 3.9487112533210917`*^9}, {3.948711482072407*^9, 674 | 3.948711493992777*^9}, {3.948728896675593*^9, 3.948728909395843*^9}}, 675 | CellLabel->"In[30]:=",ExpressionUUID->"aa933f2a-51db-4a86-8715-cd0783765465"] 676 | }, Closed]], 677 | 678 | Cell[CellGroupData[{ 679 | 680 | Cell["Stream plot of a complex number", "Subsection", 681 | CellChangeTimes->{{3.948711337333243*^9, 682 | 3.948711341470139*^9}},ExpressionUUID->"4fa8a98a-f83a-458d-9437-\ 683 | 9e0256fc484a"], 684 | 685 | Cell[BoxData[ 686 | RowBox[{"StreamPlot", "[", 687 | RowBox[{ 688 | RowBox[{"{", 689 | RowBox[{ 690 | RowBox[{"Re", "[", 691 | SuperscriptBox[ 692 | RowBox[{"(", 693 | RowBox[{"x", "+", 694 | RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], "2"], "]"}], ",", 695 | RowBox[{"Im", "[", 696 | SuperscriptBox[ 697 | RowBox[{"(", 698 | RowBox[{"x", "+", 699 | RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], "2"], "]"}]}], "}"}], ",", 700 | RowBox[{"{", 701 | RowBox[{"x", ",", 702 | RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 703 | RowBox[{"{", 704 | RowBox[{"y", ",", 705 | RowBox[{"-", "2"}], ",", "2"}], "}"}], ",", 706 | RowBox[{ 707 | "PlotLabel", 708 | "->", "\"\<\!\(\*Cell[TextData[Cell[BoxData[FormBox[RowBox[{RowBox[{\"f\",\ 709 | \"(\", \"z\", \")\"}], \"=\", 710 | SuperscriptBox[\"z\", \"2\"]}], \ 711 | TraditionalForm]],ExpressionUUID->\"fa2efc47-9895-471e-a6be-fcb1410f42f2\"]],\ 712 | ExpressionUUID->\"e06c3467-e57c-48eb-8065-87fa034f9be7\"]\)\>\""}], ",", 713 | RowBox[{"PerformanceGoal", "->", "\"\\""}], ",", 714 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]], "Input", 715 | CellChangeTimes->{{3.948707503167643*^9, 3.948707565465602*^9}, { 716 | 3.948707598165729*^9, 3.9487076614917994`*^9}, {3.94870792395779*^9, 717 | 3.948707968292075*^9}, {3.9487082065953817`*^9, 3.948708291937604*^9}, { 718 | 3.948710543229373*^9, 3.948710544813918*^9}}, 719 | CellLabel->"In[31]:=",ExpressionUUID->"8f0b3e96-e9ff-4381-ba61-e13f8a8b5689"] 720 | }, Closed]], 721 | 722 | Cell[CellGroupData[{ 723 | 724 | Cell["Complex plot", "Subsection", 725 | CellChangeTimes->{{3.9487113524257174`*^9, 726 | 3.948711354175881*^9}},ExpressionUUID->"202a3ffa-5228-4ef2-9207-\ 727 | e63e6c5940f7"], 728 | 729 | Cell[BoxData[ 730 | RowBox[{"ComplexPlot", "[", 731 | RowBox[{ 732 | SuperscriptBox["z", "2"], ",", 733 | RowBox[{"{", 734 | RowBox[{"z", ",", 735 | RowBox[{ 736 | RowBox[{"-", "2"}], "-", 737 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ",", 738 | RowBox[{"2", "+", 739 | RowBox[{"2", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", 740 | RowBox[{"PlotLegends", "->", 741 | RowBox[{"Placed", "[", 742 | RowBox[{"Automatic", ",", " ", "Left"}], "]"}]}], ",", 743 | RowBox[{"AxesLabel", "->", 744 | RowBox[{"{", 745 | RowBox[{"\"\\"", ",", "\"\\"", 746 | ",", "\"\\""}], "}"}]}], ",", 747 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]], "Input", 748 | CellChangeTimes->{{3.948710990804879*^9, 3.948710990958535*^9}, 749 | 3.948728945699971*^9, {3.948728991085359*^9, 3.948728995121861*^9}}, 750 | CellLabel->"In[32]:=",ExpressionUUID->"5bbab5ef-68bb-442e-807a-837a2b79cfe0"] 751 | }, Closed]], 752 | 753 | Cell[CellGroupData[{ 754 | 755 | Cell["3D complex plot", "Subsection", 756 | CellChangeTimes->{{3.9487113783358097`*^9, 757 | 3.9487113832079897`*^9}},ExpressionUUID->"ad1e10ca-35bf-4259-a598-\ 758 | c15d6983d8ec"], 759 | 760 | Cell[BoxData[ 761 | RowBox[{"ComplexPlot3D", "[", 762 | RowBox[{ 763 | SuperscriptBox["z", "2"], ",", 764 | RowBox[{"{", 765 | RowBox[{"z", ",", 766 | RowBox[{ 767 | RowBox[{"-", "2"}], "-", 768 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ",", 769 | RowBox[{"2", "+", 770 | RowBox[{"2", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", 771 | RowBox[{"PlotLegends", "->", 772 | RowBox[{"Placed", "[", 773 | RowBox[{"Automatic", ",", " ", "Left"}], "]"}]}], ",", 774 | RowBox[{"AxesLabel", "->", 775 | RowBox[{"{", 776 | RowBox[{"\"\\"", ",", "\"\\"", 777 | ",", "\"\\""}], "}"}]}], ",", 778 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]], "Input", 779 | CellChangeTimes->{{3.948706707620935*^9, 3.9487067655963297`*^9}, { 780 | 3.948707056938891*^9, 3.948707073987926*^9}, {3.9487072051186438`*^9, 781 | 3.948707220088271*^9}, {3.948707255284111*^9, 3.948707286495392*^9}, { 782 | 3.948707431978191*^9, 3.948707480302042*^9}, {3.948709261516107*^9, 783 | 3.948709309365879*^9}, {3.948710939674856*^9, 3.948710952077745*^9}, 784 | 3.948728942901483*^9, {3.948729003224757*^9, 3.9487290041705313`*^9}}, 785 | CellLabel->"In[33]:=",ExpressionUUID->"4a51f6c8-f6ac-4d6e-8cdc-8c1cba84440d"] 786 | }, Closed]] 787 | }, Closed]] 788 | }, Open ]] 789 | }, 790 | WindowSize->{1920, 1172}, 791 | WindowMargins->{{1792, Automatic}, {Automatic, 0}}, 792 | Magnification:>1.25 Inherited, 793 | FrontEndVersion->"14.2 for Mac OS X x86 (64-bit) (December 26, 2024)", 794 | StyleDefinitions->"Default.nb", 795 | ExpressionUUID->"758db73e-416a-4dda-8ff7-5ded4cccd628" 796 | ] 797 | (* End of Notebook Content *) 798 | 799 | (* Internal cache information *) 800 | (*CellTagsOutline 801 | CellTagsIndex->{} 802 | *) 803 | (*CellTagsIndex 804 | CellTagsIndex->{} 805 | *) 806 | (*NotebookFileOutline 807 | Notebook[{ 808 | Cell[CellGroupData[{ 809 | Cell[576, 22, 229, 4, 120, "Title",ExpressionUUID->"1ce04e14-1e2f-47ba-b751-ca726ab4557a"], 810 | Cell[808, 28, 199, 3, 65, "Subtitle",ExpressionUUID->"bd5dba02-cb2b-4859-ae1e-a57dffc8f34f"], 811 | Cell[CellGroupData[{ 812 | Cell[1032, 35, 242, 4, 82, "Section",ExpressionUUID->"f46c431e-aea3-4ea1-9b89-1787f93cc4b7"], 813 | Cell[1277, 41, 212, 6, 31, "DisplayFormulaNumbered",ExpressionUUID->"90ec4b92-5b3a-4f93-a7ab-e109392f4249"], 814 | Cell[1492, 49, 726, 17, 86, "DisplayFormulaNumbered",ExpressionUUID->"c78f3e59-1bd8-43a6-8451-771898c4ede6"], 815 | Cell[CellGroupData[{ 816 | Cell[2243, 70, 165, 3, 66, "Subsection",ExpressionUUID->"a889d8bb-5e29-41d0-9886-b8ce9d753b0c"], 817 | Cell[2411, 75, 682, 18, 42, "Text",ExpressionUUID->"6ee17c6b-8629-43ba-81d3-8787b93ecbb6"], 818 | Cell[3096, 95, 493, 14, 31, "DisplayFormulaNumbered",ExpressionUUID->"2e2bde69-21f8-4eb6-b31d-fa7a1fd6971e"], 819 | Cell[3592, 111, 252, 5, 36, "Input",ExpressionUUID->"734b99f0-3e81-4efd-acc8-fc13120eb372"], 820 | Cell[3847, 118, 1306, 32, 39, "Input",ExpressionUUID->"10c8e7d4-0ddc-4f0f-a1f8-9fc4ea439f10"] 821 | }, Closed]], 822 | Cell[CellGroupData[{ 823 | Cell[5190, 155, 155, 3, 45, "Subsection",ExpressionUUID->"d2c149d2-e2e8-42d7-a1f5-eb5919a738cb"], 824 | Cell[5348, 160, 2525, 65, 95, "Input",ExpressionUUID->"55811f37-4b56-4dbc-8d9c-b26be6f45143"], 825 | Cell[7876, 227, 4304, 103, 158, "Input",ExpressionUUID->"45018dcf-643d-447a-9cab-b9a1114abf68"] 826 | }, Closed]] 827 | }, Closed]], 828 | Cell[CellGroupData[{ 829 | Cell[12229, 336, 164, 3, 64, "Section",ExpressionUUID->"d1e63113-27ff-4d5b-8afa-ac7357c6cc11"], 830 | Cell[12396, 341, 491, 14, 31, "DisplayFormulaNumbered",ExpressionUUID->"3ade09d5-be72-42ba-9193-d054e23e04fa"], 831 | Cell[12890, 357, 399, 9, 36, "Input",ExpressionUUID->"b908dc19-4a75-4857-aa52-11fbe8bfe12a"], 832 | Cell[CellGroupData[{ 833 | Cell[13314, 370, 189, 3, 66, "Subsection",ExpressionUUID->"b229a6d3-126f-44ed-a825-0d99b0a03b32"], 834 | Cell[13506, 375, 3108, 78, 95, "Input",ExpressionUUID->"ab63d960-ee5c-4f90-85a6-b4420580a7bd"] 835 | }, Closed]], 836 | Cell[CellGroupData[{ 837 | Cell[16651, 458, 179, 3, 45, "Subsection",ExpressionUUID->"81577657-12e2-49bb-8fb5-9e51f3537e57"], 838 | Cell[16833, 463, 1373, 30, 39, "Input",ExpressionUUID->"0e80556d-da23-4b69-8e32-2cf6bc8921f9"] 839 | }, Closed]], 840 | Cell[CellGroupData[{ 841 | Cell[18243, 498, 160, 3, 45, "Subsection",ExpressionUUID->"a83c6484-a5af-4420-a3fb-8045a0c48641"], 842 | Cell[18406, 503, 829, 21, 36, "Input",ExpressionUUID->"aabb94a3-dfe3-493d-ad9a-f94c3275424e"] 843 | }, Closed]], 844 | Cell[CellGroupData[{ 845 | Cell[19272, 529, 163, 3, 45, "Subsection",ExpressionUUID->"21098d3d-267e-4fe4-8075-8c7c9616ca7f"], 846 | Cell[19438, 534, 1171, 25, 36, "Input",ExpressionUUID->"c6b14664-0064-4acd-bc9e-39cc5f5978eb"] 847 | }, Closed]] 848 | }, Closed]], 849 | Cell[CellGroupData[{ 850 | Cell[20658, 565, 175, 3, 64, "Section",ExpressionUUID->"9081f0aa-b848-483d-89ac-732f3885a2d2"], 851 | Cell[20836, 570, 492, 14, 31, "DisplayFormulaNumbered",ExpressionUUID->"3edba2ff-067a-4592-8813-fdeb35bfd5fb"], 852 | Cell[21331, 586, 411, 10, 36, "Input",ExpressionUUID->"705697d6-2b7c-423a-b71c-8b9a2a6f28c5"], 853 | Cell[CellGroupData[{ 854 | Cell[21767, 600, 185, 3, 66, "Subsection",ExpressionUUID->"72768586-42e6-400d-a77e-e3813d7e895e"], 855 | Cell[21955, 605, 2728, 69, 87, "Input",ExpressionUUID->"aa933f2a-51db-4a86-8715-cd0783765465"] 856 | }, Closed]], 857 | Cell[CellGroupData[{ 858 | Cell[24720, 679, 179, 3, 45, "Subsection",ExpressionUUID->"4fa8a98a-f83a-458d-9437-9e0256fc484a"], 859 | Cell[24902, 684, 1402, 34, 39, "Input",ExpressionUUID->"8f0b3e96-e9ff-4381-ba61-e13f8a8b5689"] 860 | }, Closed]], 861 | Cell[CellGroupData[{ 862 | Cell[26341, 723, 162, 3, 45, "Subsection",ExpressionUUID->"202a3ffa-5228-4ef2-9207-e63e6c5940f7"], 863 | Cell[26506, 728, 861, 21, 39, "Input",ExpressionUUID->"5bbab5ef-68bb-442e-807a-837a2b79cfe0"] 864 | }, Closed]], 865 | Cell[CellGroupData[{ 866 | Cell[27404, 754, 167, 3, 45, "Subsection",ExpressionUUID->"ad1e10ca-35bf-4259-a598-c15d6983d8ec"], 867 | Cell[27574, 759, 1161, 25, 39, "Input",ExpressionUUID->"4a51f6c8-f6ac-4d6e-8cdc-8c1cba84440d"] 868 | }, Closed]] 869 | }, Closed]] 870 | }, Open ]] 871 | } 872 | ] 873 | *) 874 | 875 | -------------------------------------------------------------------------------- /ProblemSet1.nb: -------------------------------------------------------------------------------- 1 | (* Content-type: application/vnd.wolfram.mathematica *) 2 | 3 | (*** Wolfram Notebook File ***) 4 | (* http://www.wolfram.com/nb *) 5 | 6 | (* CreatedBy='Wolfram 14.2' *) 7 | 8 | (*CacheID: 234*) 9 | (* Internal cache information: 10 | NotebookFileLineBreakTest 11 | NotebookFileLineBreakTest 12 | NotebookDataPosition[ 154, 7] 13 | NotebookDataLength[ 45129, 1341] 14 | NotebookOptionsPosition[ 33741, 1133] 15 | NotebookOutlinePosition[ 34136, 1149] 16 | CellTagsIndexPosition[ 34093, 1146] 17 | WindowFrame->Normal*) 18 | 19 | (* Beginning of Notebook Content *) 20 | Notebook[{ 21 | 22 | Cell[CellGroupData[{ 23 | Cell["PROBLEM SET 1", "Title", 24 | CellChangeTimes->{{3.949945142912462*^9, 25 | 3.949945144947687*^9}},ExpressionUUID->"a8fadb6c-98d0-4a9c-93e8-\ 26 | 06ba51733ec2"], 27 | 28 | Cell[CellGroupData[{ 29 | 30 | Cell["Problems", "Section", 31 | CellChangeTimes->{{3.949943194618877*^9, 32 | 3.9499431968242493`*^9}},ExpressionUUID->"8555fdcd-cc0e-49bf-9665-\ 33 | 67a0d4d90bff"], 34 | 35 | Cell[CellGroupData[{ 36 | 37 | Cell["Problem 1", "Subsubsection", 38 | CellChangeTimes->{{3.9499432009168463`*^9, 39 | 3.949943202651038*^9}},ExpressionUUID->"dc7996a3-f9e5-4d2f-969c-\ 40 | 856845c50a93"], 41 | 42 | Cell[BoxData[ 43 | RowBox[{ 44 | RowBox[{"(", 45 | RowBox[{"3", "+", 46 | RowBox[{"2", "i"}]}], ")"}], "+", 47 | RowBox[{"(", 48 | RowBox[{ 49 | RowBox[{"-", "7"}], "-", "i"}], ")"}]}]], "DisplayFormulaNumbered", 50 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 51 | 3.949943417957487*^9, 52 | 3.949943418290901*^9}},ExpressionUUID->"cc1dd8a3-1d7d-4dfe-bbc7-\ 53 | 5f37bf8b4c34"] 54 | }, Open ]], 55 | 56 | Cell[CellGroupData[{ 57 | 58 | Cell["Problem 2", "Subsubsection", 59 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 60 | 3.949943359722035*^9, 61 | 3.949943359769204*^9}},ExpressionUUID->"0a1ccf44-d5f1-4c58-8dff-\ 62 | 778f5a78bfd9"], 63 | 64 | Cell[BoxData[ 65 | RowBox[{ 66 | RowBox[{"[", 67 | RowBox[{ 68 | RowBox[{"(", 69 | RowBox[{"5", "+", 70 | RowBox[{"3", "i"}]}], ")"}], "+", 71 | RowBox[{"(", 72 | RowBox[{ 73 | RowBox[{"-", "1"}], "+", 74 | RowBox[{"2", "i"}]}], ")"}]}], "]"}], "-", 75 | RowBox[{"(", 76 | RowBox[{"7", "-", 77 | RowBox[{"5", "i"}]}], ")"}]}]], "DisplayFormulaNumbered", 78 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 79 | 3.949943262622278*^9, 80 | 3.949943281754875*^9}},ExpressionUUID->"5d1cee15-cdab-4a8e-b607-\ 81 | 06d08a12298c"] 82 | }, Open ]], 83 | 84 | Cell[CellGroupData[{ 85 | 86 | Cell["Problem 3", "Subsubsection", 87 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 88 | 3.9499433622120247`*^9, 89 | 3.9499433622439547`*^9}},ExpressionUUID->"416df8d6-d0ad-4bec-b0b5-\ 90 | e67dd6e623fa"], 91 | 92 | Cell[BoxData[ 93 | RowBox[{ 94 | RowBox[{ 95 | RowBox[{"(", 96 | RowBox[{"5", "+", 97 | RowBox[{"3", "i"}]}], ")"}], "+", 98 | RowBox[{"[", 99 | RowBox[{"(", 100 | RowBox[{ 101 | RowBox[{"-", "1"}], "+", 102 | RowBox[{"2", "i"}]}], ")"}], "]"}], "-", 103 | RowBox[{"(", 104 | RowBox[{"7", "-", 105 | RowBox[{"5", "i"}]}], ")"}]}], "]"}]], "DisplayFormulaNumbered", 106 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 107 | 3.94994329205285*^9, 108 | 3.949943297672161*^9}},ExpressionUUID->"88160995-1b19-4e27-9b95-\ 109 | 207e51f7fba5"] 110 | }, Open ]], 111 | 112 | Cell[CellGroupData[{ 113 | 114 | Cell["Problem 4", "Subsubsection", 115 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 116 | 3.9499433645074663`*^9},ExpressionUUID->"f6631c54-64a1-463f-a28b-\ 117 | 2bc5db40d3bc"], 118 | 119 | Cell[BoxData[ 120 | RowBox[{ 121 | RowBox[{"(", 122 | RowBox[{"2", "-", 123 | RowBox[{"3", "i"}]}], ")"}], 124 | RowBox[{"(", 125 | RowBox[{"4", "+", 126 | RowBox[{"2", "i"}]}], ")"}]}]], "DisplayFormulaNumbered", 127 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 128 | 3.9499433179355392`*^9, 129 | 3.9499433437549133`*^9}},ExpressionUUID->"c0e5394e-b3bc-4045-b349-\ 130 | e2da36816a5c"] 131 | }, Open ]], 132 | 133 | Cell[CellGroupData[{ 134 | 135 | Cell["Problem 5", "Subsubsection", 136 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 137 | 3.949943366226152*^9},ExpressionUUID->"67862792-6956-45b6-b943-\ 138 | 56a8a0812dcd"], 139 | 140 | Cell[BoxData[ 141 | RowBox[{ 142 | RowBox[{"(", 143 | RowBox[{"4", "+", 144 | RowBox[{"2", "i"}]}], ")"}], 145 | RowBox[{"(", 146 | RowBox[{"2", "-", 147 | RowBox[{"3", "i"}]}], ")"}]}]], "DisplayFormulaNumbered", 148 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 149 | 3.949943350526073*^9, 150 | 3.949943353189567*^9}},ExpressionUUID->"558cbb5a-5ae2-4f8d-9c84-\ 151 | 29dc3c08a732"] 152 | }, Open ]], 153 | 154 | Cell[CellGroupData[{ 155 | 156 | Cell["Problem6", "Subsubsection", 157 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 158 | 3.949943368391183*^9},ExpressionUUID->"8c4dca4f-b76e-4d42-a023-\ 159 | 4eb24853c444"], 160 | 161 | Cell[BoxData[ 162 | RowBox[{ 163 | RowBox[{"(", 164 | RowBox[{"2", "-", "i"}], ")"}], "[", 165 | RowBox[{ 166 | RowBox[{"(", 167 | RowBox[{ 168 | RowBox[{"-", "3"}], "+", 169 | RowBox[{"2", "i"}]}], ")"}], 170 | RowBox[{"(", 171 | RowBox[{"5", "-", 172 | RowBox[{"4", "i"}]}], ")"}]}], "]"}]], "DisplayFormulaNumbered", 173 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 174 | 3.949943397504896*^9, 3.949943403857596*^9}, {3.9499434468747883`*^9, 175 | 3.949943459757174*^9}},ExpressionUUID->"d3a1c73c-59bb-492e-a794-\ 176 | f3ec0ac07ca6"] 177 | }, Open ]], 178 | 179 | Cell[CellGroupData[{ 180 | 181 | Cell["Problem 7", "Subsubsection", 182 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 183 | 3.949943370991692*^9},ExpressionUUID->"9718fb12-b358-4cc4-b08b-\ 184 | 6ebd146a0204"], 185 | 186 | Cell[BoxData[ 187 | RowBox[{ 188 | RowBox[{"[", 189 | RowBox[{ 190 | RowBox[{"(", 191 | RowBox[{"2", "-", "i"}], ")"}], 192 | RowBox[{"(", 193 | RowBox[{ 194 | RowBox[{"-", "3"}], "+", 195 | RowBox[{"2", "i"}]}], ")"}]}], "]"}], 196 | RowBox[{"(", 197 | RowBox[{"5", "-", 198 | RowBox[{"4", "i"}]}], ")"}]}]], "DisplayFormulaNumbered", 199 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 200 | 3.9499434762925367`*^9, 201 | 3.94994348845936*^9}},ExpressionUUID->"74e4b438-5216-44f3-b018-\ 202 | 04cf98e7f67a"] 203 | }, Open ]], 204 | 205 | Cell[CellGroupData[{ 206 | 207 | Cell["Problem 8", "Subsubsection", 208 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 209 | 3.949943373188951*^9},ExpressionUUID->"b0759cf2-8c27-4ee4-be0c-\ 210 | 663b85c8bd1f"], 211 | 212 | Cell[BoxData[ 213 | RowBox[{ 214 | RowBox[{"(", 215 | RowBox[{ 216 | RowBox[{"-", "1"}], "+", 217 | RowBox[{"2", "i"}]}], ")"}], "[", 218 | RowBox[{ 219 | RowBox[{"(", 220 | RowBox[{"7", "-", 221 | RowBox[{"5", "i"}]}], ")"}], "+", 222 | RowBox[{"(", 223 | RowBox[{ 224 | RowBox[{"-", "3"}], "+", 225 | RowBox[{"4", "i"}]}], ")"}]}], "]"}]], "DisplayFormulaNumbered", 226 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 227 | 3.949943495478785*^9, 228 | 3.949943535611259*^9}},ExpressionUUID->"6d47f9f9-9fac-4bcc-afbc-\ 229 | 8daa2ebe6c02"] 230 | }, Open ]], 231 | 232 | Cell[CellGroupData[{ 233 | 234 | Cell["Problem 9", "Subsubsection", 235 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 236 | 3.949943375389052*^9},ExpressionUUID->"acd1108e-b7b3-461d-a94c-\ 237 | 47bedb258bc1"], 238 | 239 | Cell[TextData[{ 240 | "Let ", 241 | Cell[BoxData[ 242 | FormBox[ 243 | TemplateBox[<|"boxes" -> FormBox[ 244 | RowBox[{ 245 | SubscriptBox[ 246 | StyleBox["z", "TI"], "1"], "\[LongEqual]", "2", "+", 247 | StyleBox["i", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> 248 | "z_{1}=2+i", "state" -> "Boxes"|>, 249 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 250 | "60547f69-1213-4417-b632-434ee247a151"], 251 | ", and ", 252 | Cell[BoxData[ 253 | FormBox[ 254 | TemplateBox[<|"boxes" -> FormBox[ 255 | RowBox[{ 256 | SubscriptBox[ 257 | StyleBox["z", "TI"], "2"], "\[LongEqual]", "3", "-", "2", 258 | StyleBox["i", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> 259 | "z_{2}=3-2i", "state" -> "Boxes"|>, 260 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 261 | "385f81c2-6eba-4911-ba0b-2f4b00af9135"], 262 | ". Calculate the following." 263 | }], "Text", 264 | CellChangeTimes->{{3.949943678744689*^9, 3.949943710827426*^9}, { 265 | 3.949943775498569*^9, 266 | 3.949943832918405*^9}},ExpressionUUID->"2243755f-a459-4e01-8bc1-\ 267 | e14b09da96eb"], 268 | 269 | Cell[BoxData[ 270 | RowBox[{"|", 271 | RowBox[{ 272 | RowBox[{"3", 273 | SubscriptBox["z", "1"]}], "-", 274 | RowBox[{"4", 275 | SubscriptBox["z", "2"]}]}], "|"}]], "DisplayFormulaNumbered", 276 | CellChangeTimes->{{3.949943630981949*^9, 277 | 3.9499436397001038`*^9}},ExpressionUUID->"333c6c68-e37f-4514-8c42-\ 278 | cd703721669b"] 279 | }, Open ]], 280 | 281 | Cell[CellGroupData[{ 282 | 283 | Cell["Problem 10", "Subsubsection", 284 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 285 | 3.949943377525193*^9},ExpressionUUID->"ca421af5-c4f7-4eba-88cc-\ 286 | 21c3603ac046"], 287 | 288 | Cell[TextData[{ 289 | "Let ", 290 | Cell[BoxData[ 291 | FormBox[ 292 | TemplateBox[<|"boxes" -> FormBox[ 293 | RowBox[{ 294 | StyleBox["z", "TI"], "\[LongEqual]", "2", "+", 295 | StyleBox["i", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> 296 | "z=2+i", "state" -> "Boxes"|>, 297 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 298 | "25bd5833-e0d6-4679-a7c7-f832437d5e85"], 299 | ". Calculate the following." 300 | }], "Text", 301 | CellChangeTimes->{{3.949943842801509*^9, 3.949943854421883*^9}, { 302 | 3.9499439363543453`*^9, 303 | 3.9499439393230257`*^9}},ExpressionUUID->"653be255-70af-4626-bccd-\ 304 | 3405a4cf8fb1"], 305 | 306 | Cell[BoxData[ 307 | RowBox[{ 308 | SuperscriptBox["z", "3"], "-", 309 | RowBox[{"3", 310 | SuperscriptBox["z", "2"]}], "+", 311 | RowBox[{"4", "z"}], "-", "8"}]], "DisplayFormulaNumbered", 312 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 313 | 3.949943886970695*^9, 314 | 3.9499439545361547`*^9}},ExpressionUUID->"9a3e9852-0c9c-494c-b528-\ 315 | 6d9c94c7c68c"] 316 | }, Open ]], 317 | 318 | Cell[CellGroupData[{ 319 | 320 | Cell["Problem 11", "Subsubsection", 321 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 322 | 3.949943379222817*^9},ExpressionUUID->"a554596a-6c0a-43b1-ba3b-\ 323 | b1a5e177329c"], 324 | 325 | Cell[TextData[{ 326 | "Let ", 327 | Cell[BoxData[ 328 | FormBox[ 329 | TemplateBox[<|"boxes" -> FormBox[ 330 | RowBox[{ 331 | StyleBox["z", "TI"], "\[LongEqual]", "-", 332 | FractionBox["1", "2"], "+", 333 | FractionBox[ 334 | SqrtBox["3"], "2"], 335 | StyleBox["i", "TI"]}], TraditionalForm], "errors" -> {}, "input" -> 336 | "z=- \\frac{1}{2} + \\frac{\\sqrt{3}}{2} i", "state" -> "Boxes"|>, 337 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 338 | "254002a7-32cc-450b-b41a-974cca823b34"], 339 | ". Calculate the following." 340 | }], "Text", 341 | CellChangeTimes->{ 342 | 3.949943970867208*^9},ExpressionUUID->"b536d289-787e-498a-93bd-\ 343 | 288135036694"], 344 | 345 | Cell[BoxData[ 346 | SuperscriptBox["z", "4"]], "DisplayFormulaNumbered", 347 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 348 | 3.94994399648452*^9, 349 | 3.949943997743226*^9}},ExpressionUUID->"d833d63b-b7e5-4a88-83c3-\ 350 | ef91da241cce"] 351 | }, Open ]], 352 | 353 | Cell[CellGroupData[{ 354 | 355 | Cell["Problem 12", "Subsubsection", 356 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 357 | 3.949943381319701*^9},ExpressionUUID->"5341f231-d9d8-4ce0-bb93-\ 358 | 0ebf9708a7bf"], 359 | 360 | Cell[TextData[{ 361 | "Find ", 362 | Cell[BoxData[ 363 | FormBox[ 364 | TemplateBox[<|"boxes" -> FormBox[ 365 | StyleBox["x", "TI"], TraditionalForm], "errors" -> {}, "input" -> "x", 366 | "state" -> "Boxes"|>, 367 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 368 | "3af90263-d1e6-4cc0-afef-9f7098d258a0"], 369 | " and ", 370 | Cell[BoxData[ 371 | FormBox[ 372 | TemplateBox[<|"boxes" -> FormBox[ 373 | StyleBox["y", "TI"], TraditionalForm], "errors" -> {}, "input" -> "y", 374 | "state" -> "Boxes"|>, 375 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 376 | "c4f296b6-f2a4-472f-9f3d-fde459e94437"], 377 | " if the following holds." 378 | }], "Text", 379 | CellChangeTimes->{{3.949944015472369*^9, 380 | 3.949944028889337*^9}},ExpressionUUID->"3833e842-e3ff-4704-a687-\ 381 | 9515a2a2be73"], 382 | 383 | Cell[BoxData[ 384 | RowBox[{ 385 | RowBox[{ 386 | RowBox[{"3", "x"}], "+", 387 | RowBox[{"2", "yi"}], "-", "ix", "+", 388 | RowBox[{"5", "y"}]}], "=", 389 | RowBox[{"7", "+", 390 | RowBox[{"5", "i"}]}]}]], "DisplayFormulaNumbered", 391 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 392 | 3.949944034120305*^9, 393 | 3.9499440586749897`*^9}},ExpressionUUID->"a9f84e34-154e-448a-9890-\ 394 | 0f30c2a43ac9"] 395 | }, Open ]], 396 | 397 | Cell[CellGroupData[{ 398 | 399 | Cell["Problem 13", "Subsubsection", 400 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 401 | 3.949943382225985*^9},ExpressionUUID->"6f8bc841-731e-4a6e-84e9-\ 402 | 79d5000e1985"], 403 | 404 | Cell["Express the following complex number in polar form.", "Text", 405 | CellChangeTimes->{{3.9499440750402327`*^9, 406 | 3.94994408923834*^9}},ExpressionUUID->"e6b6ede9-ae7f-4c99-9a35-\ 407 | 514aef33fdac"], 408 | 409 | Cell[BoxData[ 410 | RowBox[{"2", "+", 411 | RowBox[{"2", 412 | SqrtBox["3"], "i"}]}]], "DisplayFormulaNumbered", 413 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 414 | 3.949944094674135*^9, 415 | 3.9499441003406076`*^9}},ExpressionUUID->"2ed269ab-cc9d-4d42-b3c3-\ 416 | 118821aeb9d5"] 417 | }, Open ]], 418 | 419 | Cell[CellGroupData[{ 420 | 421 | Cell["Problem 14", "Subsubsection", 422 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 423 | 3.949943383646349*^9},ExpressionUUID->"824b360d-e3d9-4651-bb07-\ 424 | 869663ca8476"], 425 | 426 | Cell["Express the following complex number in polar form.", "Text", 427 | CellChangeTimes->{{3.9499440750402327`*^9, 3.94994408923834*^9}, { 428 | 3.949944121641873*^9, 429 | 3.94994412260845*^9}},ExpressionUUID->"98dfa17f-da1e-4f71-9f59-\ 430 | c879fd9c6bb9"], 431 | 432 | Cell[BoxData[ 433 | RowBox[{ 434 | RowBox[{"-", "5"}], "+", 435 | RowBox[{"5", "i"}]}]], "DisplayFormulaNumbered", 436 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 437 | 3.949944120275909*^9, 438 | 3.949944130691847*^9}},ExpressionUUID->"2ebf494e-6d37-4c8e-939a-\ 439 | b3b7782ff144"] 440 | }, Open ]], 441 | 442 | Cell[CellGroupData[{ 443 | 444 | Cell["Problem 15", "Subsubsection", 445 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 446 | 3.949943386878392*^9, {3.949944171825521*^9, 447 | 3.949944171895739*^9}},ExpressionUUID->"9c03f271-05a3-4d59-bbdf-\ 448 | 661230181e14"], 449 | 450 | Cell["Express the following complex number in rectangular from", "Text", 451 | CellChangeTimes->{{3.9499441762442102`*^9, 452 | 3.9499441932428703`*^9}},ExpressionUUID->"b1934438-c8b6-44dd-a99b-\ 453 | 587015ae1b25"], 454 | 455 | Cell[BoxData[ 456 | RowBox[{"3", 457 | SuperscriptBox["e", 458 | RowBox[{ 459 | FractionBox["\[Pi]", "2"], "i"}]]}]], "DisplayFormulaNumbered", 460 | CellChangeTimes->{{3.949943216834263*^9, 3.949943228104889*^9}, { 461 | 3.949944195946199*^9, 462 | 3.949944205076579*^9}},ExpressionUUID->"ca881752-5f45-4b9c-872e-\ 463 | 650d75a56b5d"] 464 | }, Open ]] 465 | }, Open ]], 466 | 467 | Cell[CellGroupData[{ 468 | 469 | Cell["Solutions", "Section", 470 | CellChangeTimes->{{3.949944136991487*^9, 471 | 3.949944139026559*^9}},ExpressionUUID->"44d10dc0-33c1-48a9-82e7-\ 472 | 3ea2ac9fb46f"], 473 | 474 | Cell[CellGroupData[{ 475 | 476 | Cell["Problem 1", "Subsubsection", 477 | CellChangeTimes->{{3.9499432009168463`*^9, 478 | 3.949943202651038*^9}},ExpressionUUID->"f9454f94-d063-43da-9ede-\ 479 | 928c44af3195"], 480 | 481 | Cell[CellGroupData[{ 482 | 483 | Cell[BoxData[ 484 | RowBox[{ 485 | RowBox[{"(", 486 | RowBox[{"3", "+", 487 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}], "+", 488 | RowBox[{"(", 489 | RowBox[{ 490 | RowBox[{"-", "7"}], "-", "\[ImaginaryI]"}], ")"}]}]], "Input", 491 | CellChangeTimes->{{3.9499442728793983`*^9, 3.949944299428583*^9}}, 492 | CellLabel->"In[2]:=",ExpressionUUID->"83caaa0c-dcb3-4036-90a6-c36deb7b5f13"], 493 | 494 | Cell[BoxData[ 495 | RowBox[{ 496 | RowBox[{"-", "4"}], "+", "\[ImaginaryI]"}]], "Output", 497 | CellChangeTimes->{{3.9499442853681707`*^9, 3.9499442997091103`*^9}}, 498 | CellLabel->"Out[2]=",ExpressionUUID->"86a9dc29-2d77-434e-94da-207e14cf571b"] 499 | }, Open ]] 500 | }, Open ]], 501 | 502 | Cell[CellGroupData[{ 503 | 504 | Cell["Problem 2", "Subsubsection", 505 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 506 | 3.9499442313599663`*^9, 507 | 3.949944231410089*^9}},ExpressionUUID->"6fc6f7f5-7656-48b6-8bdc-\ 508 | 2a592eaf5c9e"], 509 | 510 | Cell[CellGroupData[{ 511 | 512 | Cell[BoxData[ 513 | RowBox[{ 514 | RowBox[{"(", 515 | RowBox[{ 516 | RowBox[{"(", 517 | RowBox[{"5", "+", 518 | RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", 519 | RowBox[{"(", 520 | RowBox[{ 521 | RowBox[{"-", "1"}], "+", 522 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}]}], ")"}], "-", 523 | RowBox[{"(", 524 | RowBox[{"7", "-", 525 | RowBox[{"5", " ", "\[ImaginaryI]"}]}], ")"}]}]], "Input", 526 | CellChangeTimes->{{3.9499443203648567`*^9, 3.949944346292892*^9}}, 527 | CellLabel->"In[3]:=",ExpressionUUID->"e217fdd5-6124-4e93-803d-647c799a19ce"], 528 | 529 | Cell[BoxData[ 530 | RowBox[{ 531 | RowBox[{"-", "3"}], "+", 532 | RowBox[{"10", " ", "\[ImaginaryI]"}]}]], "Output", 533 | CellChangeTimes->{3.949944346898385*^9}, 534 | CellLabel->"Out[3]=",ExpressionUUID->"4dc55068-44ef-48fd-8f2f-5b039d64e45f"] 535 | }, Open ]] 536 | }, Open ]], 537 | 538 | Cell[CellGroupData[{ 539 | 540 | Cell["Problem 3", "Subsubsection", 541 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 542 | 3.949944234194112*^9, 543 | 3.9499442342582073`*^9}},ExpressionUUID->"db05df7f-26e0-4ea1-a0bc-\ 544 | a396dfff14ea"], 545 | 546 | Cell[CellGroupData[{ 547 | 548 | Cell[BoxData[ 549 | RowBox[{ 550 | RowBox[{"(", 551 | RowBox[{"5", "+", 552 | RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", 553 | RowBox[{"(", 554 | RowBox[{ 555 | RowBox[{"(", 556 | RowBox[{ 557 | RowBox[{"-", "1"}], "+", 558 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}], "-", 559 | RowBox[{"(", 560 | RowBox[{"7", "-", 561 | RowBox[{"5", " ", "\[ImaginaryI]"}]}], ")"}]}], ")"}]}]], "Input", 562 | CellChangeTimes->{{3.9499443581114063`*^9, 3.949944364441725*^9}}, 563 | CellLabel->"In[4]:=",ExpressionUUID->"5f7922b9-3427-4752-9869-4236332a6f80"], 564 | 565 | Cell[BoxData[ 566 | RowBox[{ 567 | RowBox[{"-", "3"}], "+", 568 | RowBox[{"10", " ", "\[ImaginaryI]"}]}]], "Output", 569 | CellChangeTimes->{3.949944364718392*^9}, 570 | CellLabel->"Out[4]=",ExpressionUUID->"749cbdf6-b1b5-41e3-a71c-83c2a37042ea"] 571 | }, Open ]] 572 | }, Open ]], 573 | 574 | Cell[CellGroupData[{ 575 | 576 | Cell["Problem 4", "Subsubsection", 577 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 578 | 3.9499442365441103`*^9, 579 | 3.949944236609982*^9}},ExpressionUUID->"fd9c5b8b-d4aa-4359-b91c-\ 580 | 17e949cc509e"], 581 | 582 | Cell[CellGroupData[{ 583 | 584 | Cell[BoxData[ 585 | RowBox[{ 586 | RowBox[{"(", 587 | RowBox[{"2", "-", 588 | RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], " ", 589 | RowBox[{"(", 590 | RowBox[{"4", "+", 591 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}]}]], "Input", 592 | CellChangeTimes->{{3.949944372964209*^9, 3.949944384726418*^9}}, 593 | CellLabel->"In[5]:=",ExpressionUUID->"72a71b78-edea-48d0-8039-a667ff74ef54"], 594 | 595 | Cell[BoxData[ 596 | RowBox[{"14", "-", 597 | RowBox[{"8", " ", "\[ImaginaryI]"}]}]], "Output", 598 | CellChangeTimes->{3.949944385347418*^9}, 599 | CellLabel->"Out[5]=",ExpressionUUID->"390c824e-d307-4699-ba5a-06163a0128d7"] 600 | }, Open ]] 601 | }, Open ]], 602 | 603 | Cell[CellGroupData[{ 604 | 605 | Cell["Problem 5", "Subsubsection", 606 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 607 | 3.949944238293111*^9, 608 | 3.949944238360015*^9}},ExpressionUUID->"ecbd5693-f444-439e-9ffd-\ 609 | aeda598b9fe1"], 610 | 611 | Cell[CellGroupData[{ 612 | 613 | Cell[BoxData[ 614 | RowBox[{ 615 | RowBox[{"(", 616 | RowBox[{"4", "+", 617 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}], " ", 618 | RowBox[{"(", 619 | RowBox[{"2", "-", 620 | RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}]}]], "Input", 621 | CellChangeTimes->{{3.949944404504373*^9, 3.949944407230647*^9}}, 622 | CellLabel->"In[6]:=",ExpressionUUID->"6af3d76a-a07b-4f19-ba27-7684fb48e895"], 623 | 624 | Cell[BoxData[ 625 | RowBox[{"14", "-", 626 | RowBox[{"8", " ", "\[ImaginaryI]"}]}]], "Output", 627 | CellChangeTimes->{3.949944407485795*^9}, 628 | CellLabel->"Out[6]=",ExpressionUUID->"05c6309a-1984-4303-86ec-195d354fbe8d"] 629 | }, Open ]] 630 | }, Open ]], 631 | 632 | Cell[CellGroupData[{ 633 | 634 | Cell["Problem 6", "Subsubsection", 635 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 636 | 3.949944240844707*^9},ExpressionUUID->"55043ab3-5a94-4ec4-b54e-\ 637 | a59e4c483b29"], 638 | 639 | Cell[CellGroupData[{ 640 | 641 | Cell[BoxData[ 642 | RowBox[{ 643 | RowBox[{"(", 644 | RowBox[{"2", "-", "\[ImaginaryI]"}], ")"}], " ", 645 | RowBox[{"(", 646 | RowBox[{ 647 | RowBox[{"(", 648 | RowBox[{ 649 | RowBox[{"-", "3"}], "+", 650 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}], " ", 651 | RowBox[{"(", 652 | RowBox[{"5", "-", 653 | RowBox[{"4", " ", "\[ImaginaryI]"}]}], ")"}]}], ")"}]}]], "Input", 654 | CellChangeTimes->{{3.949944414665723*^9, 3.949944440881254*^9}}, 655 | CellLabel->"In[8]:=",ExpressionUUID->"0266ede1-7915-4349-ab1f-28262dd07960"], 656 | 657 | Cell[BoxData[ 658 | RowBox[{"8", "+", 659 | RowBox[{"51", " ", "\[ImaginaryI]"}]}]], "Output", 660 | CellChangeTimes->{{3.9499444352117023`*^9, 3.949944441428555*^9}}, 661 | CellLabel->"Out[8]=",ExpressionUUID->"89af6db7-69c6-4ab0-a44f-2c0ffa30800f"] 662 | }, Open ]] 663 | }, Open ]], 664 | 665 | Cell[CellGroupData[{ 666 | 667 | Cell["Problem 7", "Subsubsection", 668 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, { 669 | 3.949944242508218*^9, 670 | 3.94994424520688*^9}},ExpressionUUID->"d5683337-9b5e-4095-a2d4-\ 671 | a25ffabe26f7"], 672 | 673 | Cell[CellGroupData[{ 674 | 675 | Cell[BoxData[ 676 | RowBox[{ 677 | RowBox[{"(", 678 | RowBox[{ 679 | RowBox[{"(", 680 | RowBox[{"2", "-", "\[ImaginaryI]"}], ")"}], " ", 681 | RowBox[{"(", 682 | RowBox[{ 683 | RowBox[{"-", "3"}], "+", 684 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}]}], ")"}], " ", 685 | RowBox[{"(", 686 | RowBox[{"5", "-", 687 | RowBox[{"4", " ", "\[ImaginaryI]"}]}], ")"}]}]], "Input", 688 | CellChangeTimes->{{3.949944532978489*^9, 3.949944538883725*^9}}, 689 | CellLabel->"In[10]:=",ExpressionUUID->"74c8dadd-5e53-4525-a730-f02cf9df5750"], 690 | 691 | Cell[BoxData[ 692 | RowBox[{"8", "+", 693 | RowBox[{"51", " ", "\[ImaginaryI]"}]}]], "Output", 694 | CellChangeTimes->{3.9499445394050694`*^9}, 695 | CellLabel->"Out[10]=",ExpressionUUID->"74ef64ff-9d04-4102-a9af-0d7301690213"] 696 | }, Open ]] 697 | }, Open ]], 698 | 699 | Cell[CellGroupData[{ 700 | 701 | Cell["Problem 8", "Subsubsection", 702 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 703 | 3.9499442444633293`*^9},ExpressionUUID->"6d42358f-195d-4947-adac-\ 704 | bc857369f2da"], 705 | 706 | Cell[BoxData[ 707 | RowBox[{ 708 | RowBox[{"(", 709 | RowBox[{ 710 | RowBox[{"-", "1"}], "+", 711 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}], " ", 712 | RowBox[{"(", 713 | RowBox[{ 714 | RowBox[{"(", 715 | RowBox[{"7", "-", 716 | RowBox[{"5", " ", "\[ImaginaryI]"}]}], ")"}], "+", 717 | RowBox[{"(", 718 | RowBox[{ 719 | RowBox[{"-", "3"}], "+", 720 | RowBox[{"4", " ", "\[ImaginaryI]"}]}], ")"}]}], ")"}]}]], "Input", 721 | CellChangeTimes->{{3.949944455399723*^9, 3.949944477310471*^9}}, 722 | CellLabel->"In[9]:=",ExpressionUUID->"12de4c23-1b3a-48d0-967c-c21149ae5e9f"] 723 | }, Open ]], 724 | 725 | Cell[CellGroupData[{ 726 | 727 | Cell["Problem 9", "Subsubsection", 728 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 729 | 3.949944248460836*^9},ExpressionUUID->"92c517ef-52e6-4aac-baf3-\ 730 | 66c929a5aacd"], 731 | 732 | Cell[CellGroupData[{ 733 | 734 | Cell[BoxData[ 735 | RowBox[{"Arg", "[", 736 | RowBox[{ 737 | RowBox[{"3", " ", 738 | RowBox[{"(", 739 | RowBox[{"2", "+", "\[ImaginaryI]"}], ")"}]}], "-", 740 | RowBox[{"4", " ", 741 | RowBox[{"(", 742 | RowBox[{"3", "-", 743 | RowBox[{"2", " ", "\[ImaginaryI]"}]}], ")"}]}]}], "]"}]], "Input", 744 | CellChangeTimes->{{3.949944554904608*^9, 3.9499445714964657`*^9}}, 745 | CellLabel->"In[11]:=",ExpressionUUID->"500414ab-3c14-496d-97a8-b0cfb8d0a9c1"], 746 | 747 | Cell[BoxData[ 748 | RowBox[{"\[Pi]", "-", 749 | RowBox[{"ArcTan", "[", 750 | FractionBox["11", "6"], "]"}]}]], "Output", 751 | CellChangeTimes->{3.949944572122321*^9}, 752 | CellLabel->"Out[11]=",ExpressionUUID->"3153e677-6cf8-4471-a13f-35197078ef53"] 753 | }, Open ]] 754 | }, Open ]], 755 | 756 | Cell[CellGroupData[{ 757 | 758 | Cell["Problem 10", "Subsubsection", 759 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 760 | 3.949944249643873*^9},ExpressionUUID->"f1f9a178-fbe4-43ab-bf5b-\ 761 | 586448eaf125"], 762 | 763 | Cell[CellGroupData[{ 764 | 765 | Cell[BoxData[ 766 | RowBox[{ 767 | SuperscriptBox[ 768 | RowBox[{"(", 769 | RowBox[{"2", "+", "\[ImaginaryI]"}], ")"}], "3"], "-", 770 | RowBox[{"3", " ", 771 | SuperscriptBox[ 772 | RowBox[{"(", 773 | RowBox[{"2", "+", "\[ImaginaryI]"}], ")"}], "2"]}], "+", 774 | RowBox[{"4", " ", 775 | RowBox[{"(", 776 | RowBox[{"2", "+", "\[ImaginaryI]"}], ")"}]}], "-", "8"}]], "Input", 777 | CellChangeTimes->{{3.9499446451363*^9, 3.949944670452868*^9}}, 778 | CellLabel->"In[13]:=",ExpressionUUID->"ba2032df-01e4-4bb7-8fdb-7a9ce6e69aab"], 779 | 780 | Cell[BoxData[ 781 | RowBox[{ 782 | RowBox[{"-", "7"}], "+", 783 | RowBox[{"3", " ", "\[ImaginaryI]"}]}]], "Output", 784 | CellChangeTimes->{3.949944670771469*^9}, 785 | CellLabel->"Out[13]=",ExpressionUUID->"ec53ac0e-b0dc-4e0d-833f-1e07dd432f9a"] 786 | }, Open ]] 787 | }, Open ]], 788 | 789 | Cell[CellGroupData[{ 790 | 791 | Cell["Problem 11", "Subsubsection", 792 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 793 | 3.9499442512822247`*^9},ExpressionUUID->"1e177923-d114-420c-90f8-\ 794 | 02e4120c40e7"], 795 | 796 | Cell[CellGroupData[{ 797 | 798 | Cell[BoxData[ 799 | SuperscriptBox[ 800 | RowBox[{"(", 801 | RowBox[{ 802 | RowBox[{"-", 803 | FractionBox["1", "2"]}], "+", 804 | RowBox[{ 805 | FractionBox[ 806 | SqrtBox["3"], "2"], " ", "\[ImaginaryI]"}]}], ")"}], "4"]], "Input", 807 | CellChangeTimes->{{3.949944692253756*^9, 3.9499447051701927`*^9}}, 808 | CellLabel->"In[14]:=",ExpressionUUID->"8f5bd47f-35ad-424f-b3a3-cda5d88539b5"], 809 | 810 | Cell[BoxData[ 811 | SuperscriptBox[ 812 | RowBox[{"(", 813 | RowBox[{ 814 | RowBox[{"-", 815 | FractionBox["1", "2"]}], "+", 816 | FractionBox[ 817 | RowBox[{"\[ImaginaryI]", " ", 818 | SqrtBox["3"]}], "2"]}], ")"}], "4"]], "Output", 819 | CellChangeTimes->{3.949944706072548*^9}, 820 | CellLabel->"Out[14]=",ExpressionUUID->"2f604cf7-1ee3-470d-93df-a28e2b54e179"] 821 | }, Open ]] 822 | }, Open ]], 823 | 824 | Cell[CellGroupData[{ 825 | 826 | Cell["Problem 12", "Subsubsection", 827 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 828 | 3.949944252097617*^9},ExpressionUUID->"b9a7dc3f-0f4c-4ac4-8749-\ 829 | f51afe44d044"], 830 | 831 | Cell[BoxData[{ 832 | RowBox[{ 833 | RowBox[{ 834 | RowBox[{"3", "x"}], "+", 835 | RowBox[{"2", "yi"}], "-", "ix", "+", 836 | RowBox[{"5", "y"}]}], "=", 837 | RowBox[{"7", "+", 838 | RowBox[{"5", "i"}]}]}], "\[IndentingNewLine]", 839 | RowBox[{ 840 | RowBox[{ 841 | RowBox[{"(", 842 | RowBox[{ 843 | RowBox[{"3", "x"}], "+", 844 | RowBox[{"5", "y"}]}], ")"}], "+", 845 | RowBox[{"i", 846 | RowBox[{"(", 847 | RowBox[{ 848 | RowBox[{"-", "x"}], "+", 849 | RowBox[{"2", "y"}]}], ")"}]}]}], "=", 850 | RowBox[{"7", "+", 851 | RowBox[{"5", "i"}]}]}], "\[IndentingNewLine]", 852 | RowBox[{ 853 | RowBox[{ 854 | RowBox[{"3", "x"}], "+", 855 | RowBox[{"5", "y"}]}], "=", "7"}], "\[IndentingNewLine]", 856 | RowBox[{ 857 | RowBox[{ 858 | RowBox[{"-", "x"}], "+", 859 | RowBox[{"2", "y"}]}], "=", "5"}]}], "DisplayFormula", 860 | CellChangeTimes->{{3.949944743889782*^9, 861 | 3.9499447912172623`*^9}},ExpressionUUID->"8e8c3dde-c814-4aca-8e27-\ 862 | 0ec785f9cce2"], 863 | 864 | Cell[CellGroupData[{ 865 | 866 | Cell[BoxData[ 867 | RowBox[{"MatrixForm", "@", 868 | RowBox[{"RowReduce", "[", 869 | RowBox[{"{", 870 | RowBox[{ 871 | RowBox[{"{", 872 | RowBox[{"3", ",", "5", ",", "7"}], "}"}], ",", 873 | RowBox[{"{", 874 | RowBox[{ 875 | RowBox[{"-", "1"}], ",", "2", ",", "5"}], "}"}]}], "}"}], 876 | "]"}]}]], "Input", 877 | CellChangeTimes->{{3.94994479405194*^9, 3.949944817037792*^9}}, 878 | CellLabel->"In[16]:=",ExpressionUUID->"ea07a175-ae34-4cac-ae56-8ae7ef0a6ad5"], 879 | 880 | Cell[BoxData[ 881 | TagBox[ 882 | RowBox[{"(", "\[NoBreak]", GridBox[{ 883 | {"1", "0", 884 | RowBox[{"-", "1"}]}, 885 | {"0", "1", "2"} 886 | }, 887 | GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, 888 | GridBoxSpacings->{"Columns" -> { 889 | Offset[0.27999999999999997`], { 890 | Offset[0.7]}, 891 | Offset[0.27999999999999997`]}, "Rows" -> { 892 | Offset[0.2], { 893 | Offset[0.4]}, 894 | Offset[0.2]}}], "\[NoBreak]", ")"}], 895 | Function[BoxForm`e$, 896 | MatrixForm[BoxForm`e$]]]], "Output", 897 | CellChangeTimes->{{3.949944808510725*^9, 3.9499448178096533`*^9}}, 898 | CellLabel-> 899 | "Out[16]//MatrixForm=",ExpressionUUID->"52afb505-9090-4d24-bc49-\ 900 | 965ff0bc9d06"] 901 | }, Open ]], 902 | 903 | Cell[TextData[{ 904 | "We have that ", 905 | Cell[BoxData[ 906 | FormBox[ 907 | TemplateBox[<|"boxes" -> FormBox[ 908 | RowBox[{ 909 | StyleBox["x", "TI"], "\[LongEqual]", "-1"}], TraditionalForm], 910 | "errors" -> {}, "input" -> "x=-1", "state" -> "Boxes"|>, 911 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 912 | "f1600950-1146-41c4-a80a-4fbdba978462"], 913 | " and ", 914 | Cell[BoxData[ 915 | FormBox[ 916 | TemplateBox[<|"boxes" -> FormBox[ 917 | RowBox[{ 918 | StyleBox["y", "TI"], "\[LongEqual]", "2"}], TraditionalForm], 919 | "errors" -> {}, "input" -> "y=2", "state" -> "Boxes"|>, 920 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 921 | "92b19d4a-c132-42c0-bf13-5cf81fcabb26"], 922 | "." 923 | }], "Text", 924 | CellChangeTimes->{{3.9499448254375877`*^9, 925 | 3.9499448421549473`*^9}},ExpressionUUID->"2be73b2a-2c3b-48a8-a6d0-\ 926 | e203b4009480"] 927 | }, Open ]], 928 | 929 | Cell[CellGroupData[{ 930 | 931 | Cell["Problem 13", "Subsubsection", 932 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 933 | 3.949944253393111*^9},ExpressionUUID->"a91994b9-3ccb-41a1-8d4b-\ 934 | d9caf7c743ad"], 935 | 936 | Cell[CellGroupData[{ 937 | 938 | Cell[BoxData[ 939 | RowBox[{"Abs", "[", 940 | RowBox[{"2", "+", 941 | RowBox[{"2", " ", 942 | SqrtBox["3"], " ", "\[ImaginaryI]"}]}], "]"}]], "Input", 943 | CellChangeTimes->{{3.94994487068898*^9, 3.949944876900579*^9}}, 944 | CellLabel->"In[17]:=",ExpressionUUID->"cd823a1f-7db3-4109-9d71-ed9bc34f3952"], 945 | 946 | Cell[BoxData["4"], "Output", 947 | CellChangeTimes->{3.9499448774941673`*^9}, 948 | CellLabel->"Out[17]=",ExpressionUUID->"b2b2a72c-8970-423f-a212-57e591fd57b6"] 949 | }, Open ]], 950 | 951 | Cell[CellGroupData[{ 952 | 953 | Cell[BoxData[ 954 | RowBox[{"Arg", "[", 955 | RowBox[{"2", "+", 956 | RowBox[{"2", " ", 957 | SqrtBox["3"], " ", "\[ImaginaryI]"}]}], "]"}]], "Input", 958 | CellChangeTimes->{{3.949944896723111*^9, 3.949944904636422*^9}}, 959 | CellLabel->"In[18]:=",ExpressionUUID->"7655e623-c46b-464b-9119-18dafb987486"], 960 | 961 | Cell[BoxData[ 962 | FractionBox["\[Pi]", "3"]], "Output", 963 | CellChangeTimes->{3.949944905291732*^9}, 964 | CellLabel->"Out[18]=",ExpressionUUID->"4efaf284-7c48-4057-845e-55e097b9dac9"] 965 | }, Open ]], 966 | 967 | Cell[BoxData[ 968 | RowBox[{ 969 | RowBox[{ 970 | RowBox[{"2", "+", 971 | RowBox[{"2", 972 | SqrtBox["3"], "i"}]}], "=", 973 | RowBox[{"4", 974 | SuperscriptBox["e", 975 | RowBox[{"i", 976 | RowBox[{"(", 977 | RowBox[{ 978 | FractionBox["\[Pi]", "3"], "+", 979 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}]]}]}], ",", " ", 980 | RowBox[{"k", "\[Element]", "\[DoubleStruckCapitalZ]"}]}]], "DisplayFormula",\ 981 | 982 | CellChangeTimes->{{3.949944917204809*^9, 983 | 3.94994495705186*^9}},ExpressionUUID->"077820c7-3be7-433d-bd34-\ 984 | adb2fa9e2ef9"] 985 | }, Open ]], 986 | 987 | Cell[CellGroupData[{ 988 | 989 | Cell["Problem 14", "Subsubsection", 990 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 991 | 3.949944254660365*^9},ExpressionUUID->"08f129b7-ebc4-481e-a6f7-\ 992 | e656105bb550"], 993 | 994 | Cell[CellGroupData[{ 995 | 996 | Cell[BoxData[ 997 | RowBox[{"Abs", "[", 998 | RowBox[{ 999 | RowBox[{"-", "5"}], "+", 1000 | RowBox[{"5", " ", "\[ImaginaryI]"}]}], "]"}]], "Input", 1001 | CellChangeTimes->{{3.9499449805414667`*^9, 3.949944983738199*^9}}, 1002 | CellLabel->"In[19]:=",ExpressionUUID->"7161f417-15b9-439b-adab-f5ac0cc1bd10"], 1003 | 1004 | Cell[BoxData[ 1005 | RowBox[{"5", " ", 1006 | SqrtBox["2"]}]], "Output", 1007 | CellChangeTimes->{3.949944984427581*^9}, 1008 | CellLabel->"Out[19]=",ExpressionUUID->"9afaa32d-dd4f-40a9-bf8e-8f6c0b26fb60"] 1009 | }, Open ]], 1010 | 1011 | Cell[CellGroupData[{ 1012 | 1013 | Cell[BoxData[ 1014 | RowBox[{"Arg", "[", 1015 | RowBox[{ 1016 | RowBox[{"-", "5"}], "+", 1017 | RowBox[{"5", " ", "\[ImaginaryI]"}]}], "]"}]], "Input", 1018 | CellChangeTimes->{{3.949944986392355*^9, 3.94994499238792*^9}}, 1019 | CellLabel->"In[20]:=",ExpressionUUID->"1581cad3-2038-4577-ad5e-dfd00a56cc08"], 1020 | 1021 | Cell[BoxData[ 1022 | FractionBox[ 1023 | RowBox[{"3", " ", "\[Pi]"}], "4"]], "Output", 1024 | CellChangeTimes->{3.949944993041438*^9}, 1025 | CellLabel->"Out[20]=",ExpressionUUID->"9a8ad0ec-4cab-46ff-b61f-db6580e16edc"] 1026 | }, Open ]], 1027 | 1028 | Cell[BoxData[ 1029 | RowBox[{ 1030 | RowBox[{ 1031 | RowBox[{ 1032 | RowBox[{"-", "5"}], "+", 1033 | RowBox[{"5", "i"}]}], "=", 1034 | RowBox[{"5", 1035 | SqrtBox["2"], 1036 | SuperscriptBox["e", 1037 | RowBox[{"i", 1038 | RowBox[{"(", 1039 | RowBox[{ 1040 | FractionBox[ 1041 | RowBox[{"3", "\[Pi]"}], "4"], "+", 1042 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}]]}]}], ",", " ", 1043 | RowBox[{"k", "\[Element]", "\[DoubleStruckCapitalZ]"}]}]], "DisplayFormula",\ 1044 | 1045 | CellChangeTimes->{{3.949944917204809*^9, 3.94994495705186*^9}, { 1046 | 3.949945000892235*^9, 1047 | 3.949945017040283*^9}},ExpressionUUID->"8005d89c-ace6-40ee-883e-\ 1048 | 3a2bf3984260"] 1049 | }, Open ]], 1050 | 1051 | Cell[CellGroupData[{ 1052 | 1053 | Cell["Problem 15", "Subsubsection", 1054 | CellChangeTimes->{{3.9499432009168463`*^9, 3.949943202651038*^9}, 1055 | 3.949944256026388*^9},ExpressionUUID->"8eb08c18-9d95-4511-ae52-\ 1056 | 7785c8600d06"], 1057 | 1058 | Cell[BoxData[{ 1059 | RowBox[{"3", 1060 | SuperscriptBox["e", 1061 | RowBox[{ 1062 | FractionBox["\[Pi]", "2"], "i"}]]}], "\[IndentingNewLine]", 1063 | RowBox[{"\[Theta]", "=", 1064 | FractionBox["\[Pi]", "2"]}], "\[IndentingNewLine]", 1065 | RowBox[{"r", "=", "3"}], "\[IndentingNewLine]", 1066 | RowBox[{"x", "=", 1067 | RowBox[{ 1068 | RowBox[{"r", " ", "cos", 1069 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1070 | RowBox[{ 1071 | RowBox[{"3", " ", "cos", 1072 | RowBox[{"(", 1073 | FractionBox["\[Pi]", "2"], ")"}]}], "=", "0"}]}]}], "\[IndentingNewLine]", 1074 | RowBox[{"y", "=", 1075 | RowBox[{ 1076 | RowBox[{"r", " ", "sin", 1077 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1078 | RowBox[{ 1079 | RowBox[{"3", "sin", 1080 | RowBox[{"(", 1081 | FractionBox["\[Pi]", "2"], ")"}]}], "=", "3"}]}]}], "\[IndentingNewLine]", 1082 | RowBox[{"3", "i"}]}], "DisplayFormula", 1083 | CellChangeTimes->{{3.949945045693376*^9, 1084 | 3.9499451163267193`*^9}},ExpressionUUID->"87cecdc5-a64d-4003-ae32-\ 1085 | e7de31fb0aec"], 1086 | 1087 | Cell[CellGroupData[{ 1088 | 1089 | Cell[BoxData[ 1090 | RowBox[{ 1091 | RowBox[{"Arg", "[", 1092 | RowBox[{ 1093 | RowBox[{"(", 1094 | RowBox[{"3", "+", 1095 | RowBox[{"4", " ", "\[ImaginaryI]"}]}], ")"}], " ", 1096 | RowBox[{"(", 1097 | RowBox[{"2", "-", "\[ImaginaryI]"}], ")"}]}], "]"}], "//", 1098 | "N"}]], "Input", 1099 | CellChangeTimes->{{3.94994564264073*^9, 3.949945667791024*^9}, { 1100 | 3.949945698776037*^9, 3.949945701676002*^9}, {3.949945734097744*^9, 1101 | 3.949945774529608*^9}}, 1102 | CellLabel->"In[30]:=",ExpressionUUID->"91c98515-395c-4f0c-a85d-7b6aeed7d075"], 1103 | 1104 | Cell[BoxData["0.46364760900080615`"], "Output", 1105 | CellChangeTimes->{{3.949945659275569*^9, 3.949945668135737*^9}, 1106 | 3.949945702390036*^9, {3.949945744905656*^9, 3.949945775132778*^9}}, 1107 | CellLabel->"Out[30]=",ExpressionUUID->"11be3ad6-bf76-4bfc-a84e-0b9c961039f9"] 1108 | }, Open ]], 1109 | 1110 | Cell[CellGroupData[{ 1111 | 1112 | Cell[BoxData[ 1113 | RowBox[{ 1114 | RowBox[{ 1115 | RowBox[{"Arg", "[", 1116 | RowBox[{"3", "+", 1117 | RowBox[{"4", " ", "\[ImaginaryI]"}]}], "]"}], " ", 1118 | RowBox[{"Arg", "[", 1119 | RowBox[{"2", "-", "\[ImaginaryI]"}], "]"}]}], "//", "N"}]], "Input", 1120 | CellChangeTimes->{{3.94994566960881*^9, 3.949945705574684*^9}, { 1121 | 3.9499457523751698`*^9, 3.9499457816801662`*^9}}, 1122 | CellLabel->"In[31]:=",ExpressionUUID->"3400dc14-883e-405b-885e-7519fa2d2558"], 1123 | 1124 | Cell[BoxData[ 1125 | RowBox[{"-", "0.4299382106643288`"}]], "Output", 1126 | CellChangeTimes->{{3.9499456809595203`*^9, 3.949945706063843*^9}, { 1127 | 3.949945755660982*^9, 3.9499457819977703`*^9}}, 1128 | CellLabel->"Out[31]=",ExpressionUUID->"be603558-bae4-4767-8934-5793d43a227f"] 1129 | }, Open ]] 1130 | }, Open ]] 1131 | }, Open ]] 1132 | }, Open ]] 1133 | }, 1134 | WindowSize->{921, 911}, 1135 | WindowMargins->{{4, Automatic}, {Automatic, 4}}, 1136 | FrontEndVersion->"14.2 for Mac OS X x86 (64-bit) (December 26, 2024)", 1137 | StyleDefinitions->"Default.nb", 1138 | ExpressionUUID->"87419ff0-ec36-4270-9df2-a592b153180d" 1139 | ] 1140 | (* End of Notebook Content *) 1141 | 1142 | (* Internal cache information *) 1143 | (*CellTagsOutline 1144 | CellTagsIndex->{} 1145 | *) 1146 | (*CellTagsIndex 1147 | CellTagsIndex->{} 1148 | *) 1149 | (*NotebookFileOutline 1150 | Notebook[{ 1151 | Cell[CellGroupData[{ 1152 | Cell[576, 22, 156, 3, 96, "Title",ExpressionUUID->"a8fadb6c-98d0-4a9c-93e8-06ba51733ec2"], 1153 | Cell[CellGroupData[{ 1154 | Cell[757, 29, 155, 3, 66, "Section",ExpressionUUID->"8555fdcd-cc0e-49bf-9665-67a0d4d90bff"], 1155 | Cell[CellGroupData[{ 1156 | Cell[937, 36, 162, 3, 43, "Subsubsection",ExpressionUUID->"dc7996a3-f9e5-4d2f-969c-856845c50a93"], 1157 | Cell[1102, 41, 375, 11, 25, "DisplayFormulaNumbered",ExpressionUUID->"cc1dd8a3-1d7d-4dfe-bbc7-5f37bf8b4c34"] 1158 | }, Open ]], 1159 | Cell[CellGroupData[{ 1160 | Cell[1514, 57, 211, 4, 43, "Subsubsection",ExpressionUUID->"0a1ccf44-d5f1-4c58-8dff-778f5a78bfd9"], 1161 | Cell[1728, 63, 523, 17, 25, "DisplayFormulaNumbered",ExpressionUUID->"5d1cee15-cdab-4a8e-b607-06d08a12298c"] 1162 | }, Open ]], 1163 | Cell[CellGroupData[{ 1164 | Cell[2288, 85, 215, 4, 43, "Subsubsection",ExpressionUUID->"416df8d6-d0ad-4bec-b0b5-e67dd6e623fa"], 1165 | Cell[2506, 91, 527, 17, 25, "DisplayFormulaNumbered",ExpressionUUID->"88160995-1b19-4e27-9b95-207e51f7fba5"] 1166 | }, Open ]], 1167 | Cell[CellGroupData[{ 1168 | Cell[3070, 113, 187, 3, 43, "Subsubsection",ExpressionUUID->"f6631c54-64a1-463f-a28b-2bc5db40d3bc"], 1169 | Cell[3260, 118, 374, 11, 25, "DisplayFormulaNumbered",ExpressionUUID->"c0e5394e-b3bc-4045-b349-e2da36816a5c"] 1170 | }, Open ]], 1171 | Cell[CellGroupData[{ 1172 | Cell[3671, 134, 185, 3, 43, "Subsubsection",ExpressionUUID->"67862792-6956-45b6-b943-56a8a0812dcd"], 1173 | Cell[3859, 139, 370, 11, 25, "DisplayFormulaNumbered",ExpressionUUID->"558cbb5a-5ae2-4f8d-9c84-29dc3c08a732"] 1174 | }, Open ]], 1175 | Cell[CellGroupData[{ 1176 | Cell[4266, 155, 184, 3, 43, "Subsubsection",ExpressionUUID->"8c4dca4f-b76e-4d42-a023-4eb24853c444"], 1177 | Cell[4453, 160, 520, 15, 25, "DisplayFormulaNumbered",ExpressionUUID->"d3a1c73c-59bb-492e-a794-f3ec0ac07ca6"] 1178 | }, Open ]], 1179 | Cell[CellGroupData[{ 1180 | Cell[5010, 180, 185, 3, 43, "Subsubsection",ExpressionUUID->"9718fb12-b358-4cc4-b08b-6ebd146a0204"], 1181 | Cell[5198, 185, 492, 16, 25, "DisplayFormulaNumbered",ExpressionUUID->"74e4b438-5216-44f3-b018-04cf98e7f67a"] 1182 | }, Open ]], 1183 | Cell[CellGroupData[{ 1184 | Cell[5727, 206, 185, 3, 43, "Subsubsection",ExpressionUUID->"b0759cf2-8c27-4ee4-be0c-663b85c8bd1f"], 1185 | Cell[5915, 211, 517, 17, 25, "DisplayFormulaNumbered",ExpressionUUID->"6d47f9f9-9fac-4bcc-afbc-8daa2ebe6c02"] 1186 | }, Open ]], 1187 | Cell[CellGroupData[{ 1188 | Cell[6469, 233, 185, 3, 43, "Subsubsection",ExpressionUUID->"acd1108e-b7b3-461d-a94c-47bedb258bc1"], 1189 | Cell[6657, 238, 1040, 28, 33, "Text",ExpressionUUID->"2243755f-a459-4e01-8bc1-e14b09da96eb"], 1190 | Cell[7700, 268, 303, 9, 25, "DisplayFormulaNumbered",ExpressionUUID->"333c6c68-e37f-4514-8c42-cd703721669b"] 1191 | }, Open ]], 1192 | Cell[CellGroupData[{ 1193 | Cell[8040, 282, 186, 3, 43, "Subsubsection",ExpressionUUID->"ca421af5-c4f7-4eba-88cc-21c3603ac046"], 1194 | Cell[8229, 287, 602, 16, 33, "Text",ExpressionUUID->"653be255-70af-4626-bccd-3405a4cf8fb1"], 1195 | Cell[8834, 305, 348, 9, 25, "DisplayFormulaNumbered",ExpressionUUID->"9a3e9852-0c9c-494c-b528-6d9c94c7c68c"] 1196 | }, Open ]], 1197 | Cell[CellGroupData[{ 1198 | Cell[9219, 319, 186, 3, 43, "Subsubsection",ExpressionUUID->"a554596a-6c0a-43b1-ba3b-b1a5e177329c"], 1199 | Cell[9408, 324, 647, 18, 54, "Text",ExpressionUUID->"b536d289-787e-498a-93bd-288135036694"], 1200 | Cell[10058, 344, 241, 5, 25, "DisplayFormulaNumbered",ExpressionUUID->"d833d63b-b7e5-4a88-83c3-ef91da241cce"] 1201 | }, Open ]], 1202 | Cell[CellGroupData[{ 1203 | Cell[10336, 354, 186, 3, 43, "Subsubsection",ExpressionUUID->"5341f231-d9d8-4ce0-bb93-0ebf9708a7bf"], 1204 | Cell[10525, 359, 753, 21, 33, "Text",ExpressionUUID->"3833e842-e3ff-4704-a687-9515a2a2be73"], 1205 | Cell[11281, 382, 387, 11, 25, "DisplayFormulaNumbered",ExpressionUUID->"a9f84e34-154e-448a-9890-0f30c2a43ac9"] 1206 | }, Open ]], 1207 | Cell[CellGroupData[{ 1208 | Cell[11705, 398, 186, 3, 43, "Subsubsection",ExpressionUUID->"6f8bc841-731e-4a6e-84e9-79d5000e1985"], 1209 | Cell[11894, 403, 194, 3, 33, "Text",ExpressionUUID->"e6b6ede9-ae7f-4c99-9a35-514aef33fdac"], 1210 | Cell[12091, 408, 279, 7, 25, "DisplayFormulaNumbered",ExpressionUUID->"2ed269ab-cc9d-4d42-b3c3-118821aeb9d5"] 1211 | }, Open ]], 1212 | Cell[CellGroupData[{ 1213 | Cell[12407, 420, 186, 3, 43, "Subsubsection",ExpressionUUID->"824b360d-e3d9-4651-bb07-869663ca8476"], 1214 | Cell[12596, 425, 242, 4, 33, "Text",ExpressionUUID->"98dfa17f-da1e-4f71-9f59-c879fd9c6bb9"], 1215 | Cell[12841, 431, 277, 7, 25, "DisplayFormulaNumbered",ExpressionUUID->"2ebf494e-6d37-4c8e-939a-b3b7782ff144"] 1216 | }, Open ]], 1217 | Cell[CellGroupData[{ 1218 | Cell[13155, 443, 236, 4, 43, "Subsubsection",ExpressionUUID->"9c03f271-05a3-4d59-bbdf-661230181e14"], 1219 | Cell[13394, 449, 202, 3, 33, "Text",ExpressionUUID->"b1934438-c8b6-44dd-a99b-587015ae1b25"], 1220 | Cell[13599, 454, 306, 8, 28, "DisplayFormulaNumbered",ExpressionUUID->"ca881752-5f45-4b9c-872e-650d75a56b5d"] 1221 | }, Open ]] 1222 | }, Open ]], 1223 | Cell[CellGroupData[{ 1224 | Cell[13954, 468, 154, 3, 52, "Section",ExpressionUUID->"44d10dc0-33c1-48a9-82e7-3ea2ac9fb46f"], 1225 | Cell[CellGroupData[{ 1226 | Cell[14133, 475, 162, 3, 43, "Subsubsection",ExpressionUUID->"f9454f94-d063-43da-9ede-928c44af3195"], 1227 | Cell[CellGroupData[{ 1228 | Cell[14320, 482, 358, 9, 29, "Input",ExpressionUUID->"83caaa0c-dcb3-4036-90a6-c36deb7b5f13"], 1229 | Cell[14681, 493, 228, 4, 33, "Output",ExpressionUUID->"86a9dc29-2d77-434e-94da-207e14cf571b"] 1230 | }, Open ]] 1231 | }, Open ]], 1232 | Cell[CellGroupData[{ 1233 | Cell[14958, 503, 213, 4, 43, "Subsubsection",ExpressionUUID->"6fc6f7f5-7656-48b6-8bdc-2a592eaf5c9e"], 1234 | Cell[CellGroupData[{ 1235 | Cell[15196, 511, 528, 15, 29, "Input",ExpressionUUID->"e217fdd5-6124-4e93-803d-647c799a19ce"], 1236 | Cell[15727, 528, 224, 5, 33, "Output",ExpressionUUID->"4dc55068-44ef-48fd-8f2f-5b039d64e45f"] 1237 | }, Open ]] 1238 | }, Open ]], 1239 | Cell[CellGroupData[{ 1240 | Cell[16000, 539, 213, 4, 43, "Subsubsection",ExpressionUUID->"db05df7f-26e0-4ea1-a0bc-a396dfff14ea"], 1241 | Cell[CellGroupData[{ 1242 | Cell[16238, 547, 528, 15, 29, "Input",ExpressionUUID->"5f7922b9-3427-4752-9869-4236332a6f80"], 1243 | Cell[16769, 564, 224, 5, 33, "Output",ExpressionUUID->"749cbdf6-b1b5-41e3-a71c-83c2a37042ea"] 1244 | }, Open ]] 1245 | }, Open ]], 1246 | Cell[CellGroupData[{ 1247 | Cell[17042, 575, 213, 4, 43, "Subsubsection",ExpressionUUID->"fd9c5b8b-d4aa-4359-b91c-17e949cc509e"], 1248 | Cell[CellGroupData[{ 1249 | Cell[17280, 583, 361, 9, 29, "Input",ExpressionUUID->"72a71b78-edea-48d0-8039-a667ff74ef54"], 1250 | Cell[17644, 594, 206, 4, 33, "Output",ExpressionUUID->"390c824e-d307-4699-ba5a-06163a0128d7"] 1251 | }, Open ]] 1252 | }, Open ]], 1253 | Cell[CellGroupData[{ 1254 | Cell[17899, 604, 211, 4, 43, "Subsubsection",ExpressionUUID->"ecbd5693-f444-439e-9ffd-aeda598b9fe1"], 1255 | Cell[CellGroupData[{ 1256 | Cell[18135, 612, 361, 9, 29, "Input",ExpressionUUID->"6af3d76a-a07b-4f19-ba27-7684fb48e895"], 1257 | Cell[18499, 623, 206, 4, 33, "Output",ExpressionUUID->"05c6309a-1984-4303-86ec-195d354fbe8d"] 1258 | }, Open ]] 1259 | }, Open ]], 1260 | Cell[CellGroupData[{ 1261 | Cell[18754, 633, 185, 3, 43, "Subsubsection",ExpressionUUID->"55043ab3-5a94-4ec4-b54e-a59e4c483b29"], 1262 | Cell[CellGroupData[{ 1263 | Cell[18964, 640, 501, 14, 29, "Input",ExpressionUUID->"0266ede1-7915-4349-ab1f-28262dd07960"], 1264 | Cell[19468, 656, 232, 4, 33, "Output",ExpressionUUID->"89af6db7-69c6-4ab0-a44f-2c0ffa30800f"] 1265 | }, Open ]] 1266 | }, Open ]], 1267 | Cell[CellGroupData[{ 1268 | Cell[19749, 666, 210, 4, 43, "Subsubsection",ExpressionUUID->"d5683337-9b5e-4095-a2d4-a25ffabe26f7"], 1269 | Cell[CellGroupData[{ 1270 | Cell[19984, 674, 500, 14, 29, "Input",ExpressionUUID->"74c8dadd-5e53-4525-a730-f02cf9df5750"], 1271 | Cell[20487, 690, 209, 4, 33, "Output",ExpressionUUID->"74ef64ff-9d04-4102-a9af-0d7301690213"] 1272 | }, Open ]] 1273 | }, Open ]], 1274 | Cell[CellGroupData[{ 1275 | Cell[20745, 700, 187, 3, 43, "Subsubsection",ExpressionUUID->"6d42358f-195d-4947-adac-bc857369f2da"], 1276 | Cell[20935, 705, 546, 16, 29, "Input",ExpressionUUID->"12de4c23-1b3a-48d0-967c-c21149ae5e9f"] 1277 | }, Open ]], 1278 | Cell[CellGroupData[{ 1279 | Cell[21518, 726, 185, 3, 43, "Subsubsection",ExpressionUUID->"92c517ef-52e6-4aac-baf3-66c929a5aacd"], 1280 | Cell[CellGroupData[{ 1281 | Cell[21728, 733, 427, 11, 29, "Input",ExpressionUUID->"500414ab-3c14-496d-97a8-b0cfb8d0a9c1"], 1282 | Cell[22158, 746, 231, 5, 46, "Output",ExpressionUUID->"3153e677-6cf8-4471-a13f-35197078ef53"] 1283 | }, Open ]] 1284 | }, Open ]], 1285 | Cell[CellGroupData[{ 1286 | Cell[22438, 757, 186, 3, 43, "Subsubsection",ExpressionUUID->"f1f9a178-fbe4-43ab-bf5b-586448eaf125"], 1287 | Cell[CellGroupData[{ 1288 | Cell[22649, 764, 492, 13, 29, "Input",ExpressionUUID->"ba2032df-01e4-4bb7-8fdb-7a9ce6e69aab"], 1289 | Cell[23144, 779, 224, 5, 33, "Output",ExpressionUUID->"ec53ac0e-b0dc-4e0d-833f-1e07dd432f9a"] 1290 | }, Open ]] 1291 | }, Open ]], 1292 | Cell[CellGroupData[{ 1293 | Cell[23417, 790, 188, 3, 43, "Subsubsection",ExpressionUUID->"1e177923-d114-420c-90f8-02e4120c40e7"], 1294 | Cell[CellGroupData[{ 1295 | Cell[23630, 797, 365, 10, 53, "Input",ExpressionUUID->"8f5bd47f-35ad-424f-b3a3-cda5d88539b5"], 1296 | Cell[23998, 809, 340, 10, 54, "Output",ExpressionUUID->"2f604cf7-1ee3-470d-93df-a28e2b54e179"] 1297 | }, Open ]] 1298 | }, Open ]], 1299 | Cell[CellGroupData[{ 1300 | Cell[24387, 825, 186, 3, 43, "Subsubsection",ExpressionUUID->"b9a7dc3f-0f4c-4ac4-8749-f51afe44d044"], 1301 | Cell[24576, 830, 874, 31, 91, "DisplayFormula",ExpressionUUID->"8e8c3dde-c814-4aca-8e27-0ec785f9cce2"], 1302 | Cell[CellGroupData[{ 1303 | Cell[25475, 865, 440, 12, 29, "Input",ExpressionUUID->"ea07a175-ae34-4cac-ae56-8ae7ef0a6ad5"], 1304 | Cell[25918, 879, 678, 20, 59, "Output",ExpressionUUID->"52afb505-9090-4d24-bc49-965ff0bc9d06"] 1305 | }, Open ]], 1306 | Cell[26611, 902, 830, 23, 33, "Text",ExpressionUUID->"2be73b2a-2c3b-48a8-a6d0-e203b4009480"] 1307 | }, Open ]], 1308 | Cell[CellGroupData[{ 1309 | Cell[27478, 930, 186, 3, 43, "Subsubsection",ExpressionUUID->"a91994b9-3ccb-41a1-8d4b-d9caf7c743ad"], 1310 | Cell[CellGroupData[{ 1311 | Cell[27689, 937, 283, 6, 34, "Input",ExpressionUUID->"cd823a1f-7db3-4109-9d71-ed9bc34f3952"], 1312 | Cell[27975, 945, 151, 2, 33, "Output",ExpressionUUID->"b2b2a72c-8970-423f-a212-57e591fd57b6"] 1313 | }, Open ]], 1314 | Cell[CellGroupData[{ 1315 | Cell[28163, 952, 284, 6, 34, "Input",ExpressionUUID->"7655e623-c46b-464b-9119-18dafb987486"], 1316 | Cell[28450, 960, 173, 3, 43, "Output",ExpressionUUID->"4efaf284-7c48-4057-845e-55e097b9dac9"] 1317 | }, Open ]], 1318 | Cell[28638, 966, 521, 17, 28, "DisplayFormula",ExpressionUUID->"077820c7-3be7-433d-bd34-adb2fa9e2ef9"] 1319 | }, Open ]], 1320 | Cell[CellGroupData[{ 1321 | Cell[29196, 988, 186, 3, 43, "Subsubsection",ExpressionUUID->"08f129b7-ebc4-481e-a6f7-e656105bb550"], 1322 | Cell[CellGroupData[{ 1323 | Cell[29407, 995, 281, 6, 29, "Input",ExpressionUUID->"7161f417-15b9-439b-adab-f5ac0cc1bd10"], 1324 | Cell[29691, 1003, 183, 4, 33, "Output",ExpressionUUID->"9afaa32d-dd4f-40a9-bf8e-8f6c0b26fb60"] 1325 | }, Open ]], 1326 | Cell[CellGroupData[{ 1327 | Cell[29911, 1012, 278, 6, 29, "Input",ExpressionUUID->"1581cad3-2038-4577-ad5e-dfd00a56cc08"], 1328 | Cell[30192, 1020, 196, 4, 46, "Output",ExpressionUUID->"9a8ad0ec-4cab-46ff-b61f-db6580e16edc"] 1329 | }, Open ]], 1330 | Cell[30403, 1027, 614, 20, 30, "DisplayFormula",ExpressionUUID->"8005d89c-ace6-40ee-883e-3a2bf3984260"] 1331 | }, Open ]], 1332 | Cell[CellGroupData[{ 1333 | Cell[31054, 1052, 186, 3, 43, "Subsubsection",ExpressionUUID->"8eb08c18-9d95-4511-ae52-7785c8600d06"], 1334 | Cell[31243, 1057, 919, 27, 176, "DisplayFormula",ExpressionUUID->"87cecdc5-a64d-4003-ae32-e7de31fb0aec"], 1335 | Cell[CellGroupData[{ 1336 | Cell[32187, 1088, 502, 13, 29, "Input",ExpressionUUID->"91c98515-395c-4f0c-a85d-7b6aeed7d075"], 1337 | Cell[32692, 1103, 264, 3, 33, "Output",ExpressionUUID->"11be3ad6-bf76-4bfc-a84e-0b9c961039f9"] 1338 | }, Open ]], 1339 | Cell[CellGroupData[{ 1340 | Cell[32993, 1111, 432, 10, 29, "Input",ExpressionUUID->"3400dc14-883e-405b-885e-7519fa2d2558"], 1341 | Cell[33428, 1123, 261, 4, 52, "Output",ExpressionUUID->"be603558-bae4-4767-8934-5793d43a227f"] 1342 | }, Open ]] 1343 | }, Open ]] 1344 | }, Open ]] 1345 | }, Open ]] 1346 | } 1347 | ] 1348 | *) 1349 | 1350 | -------------------------------------------------------------------------------- /DerivingThePolarFormOfComplexNumbers.nb: -------------------------------------------------------------------------------- 1 | (* Content-type: application/vnd.wolfram.mathematica *) 2 | 3 | (*** Wolfram Notebook File ***) 4 | (* http://www.wolfram.com/nb *) 5 | 6 | (* CreatedBy='Wolfram 14.2' *) 7 | 8 | (*CacheID: 234*) 9 | (* Internal cache information: 10 | NotebookFileLineBreakTest 11 | NotebookFileLineBreakTest 12 | NotebookDataPosition[ 154, 7] 13 | NotebookDataLength[ 67567, 1889] 14 | NotebookOptionsPosition[ 60810, 1783] 15 | NotebookOutlinePosition[ 61240, 1800] 16 | CellTagsIndexPosition[ 61197, 1797] 17 | WindowFrame->Normal*) 18 | 19 | (* Beginning of Notebook Content *) 20 | Notebook[{ 21 | 22 | Cell[CellGroupData[{ 23 | Cell["DERIVING THE EULER FORM", "Title", 24 | CellChangeTimes->{{3.9493962429678707`*^9, 3.9493962468946447`*^9}, 25 | 3.9493993705026093`*^9, {3.949429469477277*^9, 26 | 3.9494294703786583`*^9}},ExpressionUUID->"6301cd3f-0fd2-429a-9871-\ 27 | db5dacb69af2"], 28 | 29 | Cell[CellGroupData[{ 30 | 31 | Cell["Euler form of a complex number", "Section", 32 | CellChangeTimes->{{3.949396284858341*^9, 3.94939629351121*^9}, { 33 | 3.949429474095142*^9, 34 | 3.949429475023655*^9}},ExpressionUUID->"386b3412-cb3f-4c65-80ca-\ 35 | 4255d3a3917e"], 36 | 37 | Cell[BoxData[{ 38 | RowBox[{"z", "=", 39 | RowBox[{ 40 | RowBox[{"r", " ", "[", 41 | RowBox[{ 42 | RowBox[{"cos", 43 | RowBox[{"(", "\[Theta]", ")"}]}], "+", 44 | RowBox[{"i", " ", "sin", 45 | RowBox[{"(", "\[Theta]", ")"}]}]}], "]"}], "=", 46 | RowBox[{"r", " ", 47 | SuperscriptBox["e", "i\[Theta]"]}]}]}], "\[IndentingNewLine]", 48 | RowBox[{"z", "=", 49 | RowBox[{ 50 | RowBox[{"r", " ", "[", 51 | RowBox[{ 52 | RowBox[{"cos", 53 | RowBox[{"(", 54 | RowBox[{"\[Theta]", "+", 55 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}], "+", 56 | RowBox[{"i", " ", "sin", 57 | RowBox[{"(", 58 | RowBox[{"\[Theta]", "+", 59 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}]}], "]"}], "=", 60 | RowBox[{"r", " ", 61 | SuperscriptBox["e", 62 | RowBox[{"i", 63 | RowBox[{"(", 64 | RowBox[{"\[Theta]", "+", 65 | RowBox[{"2", "\[Pi]", " ", "k"}]}], 66 | ")"}]}]]}]}]}]}], "DisplayFormulaNumbered", 67 | CellChangeTimes->{{3.949396307179337*^9, 3.94939635754488*^9}, { 68 | 3.949398707757782*^9, 3.949398749795788*^9}, {3.949429527210341*^9, 69 | 3.9494295446564903`*^9}},ExpressionUUID->"f8d80c33-6b09-4cd3-9fef-\ 70 | 6718ca7fbd25"] 71 | }, Open ]], 72 | 73 | Cell[CellGroupData[{ 74 | 75 | Cell["Taylor series expansion of a function", "Section", 76 | CellChangeTimes->{{3.949396262790387*^9, 77 | 3.94939627159896*^9}},ExpressionUUID->"08fe1613-aa92-4f93-b47c-\ 78 | fbef6d46e9a6"], 79 | 80 | Cell[TextData[{ 81 | "The Taylor (power) series expansion of a function ", 82 | Cell[BoxData[ 83 | FormBox[ 84 | TemplateBox[<|"boxes" -> FormBox[ 85 | RowBox[{ 86 | StyleBox["f", "TI"], "(", "\[Theta]", ")"}], TraditionalForm], 87 | "errors" -> {}, "input" -> "f \\left( \\theta \\right)", "state" -> 88 | "Boxes"|>, 89 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 90 | "5521e48c-b908-407f-bc89-3b00df414131"], 91 | " that is infinitely differentiable at a real or complex number ", 92 | Cell[BoxData[ 93 | FormBox["a", TraditionalForm]],ExpressionUUID-> 94 | "5e5b2c60-d340-4ed8-bb8e-ece9d65e9e6a"], 95 | " is given in (2)." 96 | }], "Text", 97 | CellChangeTimes->{{3.9493964472838182`*^9, 3.949396473776319*^9}, { 98 | 3.949396520439193*^9, 3.9493965284699574`*^9}, {3.949399264962266*^9, 99 | 3.949399267432151*^9}, {3.949425648157104*^9, 3.949425679037366*^9}, { 100 | 3.9494257337104187`*^9, 101 | 3.949425738476205*^9}},ExpressionUUID->"9bccb5e1-436c-4383-b408-\ 102 | e07656fdf6ef"], 103 | 104 | Cell[BoxData[ 105 | RowBox[{ 106 | RowBox[{"f", 107 | RowBox[{"(", "a", ")"}]}], "=", 108 | RowBox[{ 109 | SubscriptBox["lim", 110 | RowBox[{"n", "->", "\[Infinity]"}]], 111 | RowBox[{ 112 | UnderoverscriptBox["\[Sum]", 113 | RowBox[{"j", "=", "0"}], "n"], 114 | RowBox[{ 115 | FractionBox[ 116 | RowBox[{ 117 | SuperscriptBox["f", 118 | RowBox[{"(", "j", ")"}]], 119 | RowBox[{"(", "a", ")"}]}], 120 | RowBox[{"j", "!"}]], 121 | SuperscriptBox[ 122 | RowBox[{"(", 123 | RowBox[{"\[Theta]", "-", "a"}], ")"}], 124 | "j"]}]}]}]}]], "DisplayFormulaNumbered", 125 | CellChangeTimes->{{3.9469112784188538`*^9, 3.946911359464532*^9}, { 126 | 3.946911390054572*^9, 3.9469114301328583`*^9}, {3.946911488918517*^9, 127 | 3.9469115622648907`*^9}, {3.949396401318059*^9, 3.949396442655059*^9}, { 128 | 3.949396509605586*^9, 3.949396510047324*^9}, {3.9493989445415707`*^9, 129 | 3.949398951943433*^9}, {3.9493991960181427`*^9, 3.949399204594284*^9}, { 130 | 3.949425547203595*^9, 3.949425549301597*^9}, {3.949425688351492*^9, 131 | 3.9494256884033823`*^9}},ExpressionUUID->"2a2029d9-3573-415d-bde0-\ 132 | 45d3680b5777"] 133 | }, Open ]], 134 | 135 | Cell[CellGroupData[{ 136 | 137 | Cell["Maclaurin series expansion", "Section", 138 | CellChangeTimes->{{3.9493964822317657`*^9, 139 | 3.949396487647344*^9}},ExpressionUUID->"7cd5c749-8cfa-4366-a774-\ 140 | 5b010afbe073"], 141 | 142 | Cell[TextData[{ 143 | "The Taylor series expansion of a function ", 144 | Cell[BoxData[ 145 | FormBox[ 146 | TemplateBox[<|"boxes" -> FormBox[ 147 | RowBox[{ 148 | StyleBox["f", "TI"], "(", "\[Theta]", ")"}], TraditionalForm], 149 | "errors" -> {}, "input" -> "f \\left( \\theta \\right)", "state" -> 150 | "Boxes"|>, 151 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 152 | "d8a31956-a822-468a-97b0-17eabe491952"], 153 | " at ", 154 | Cell[BoxData[ 155 | FormBox[ 156 | TemplateBox[<|"boxes" -> FormBox[ 157 | RowBox[{ 158 | StyleBox["a", "TI"], "\[LongEqual]", "0"}], TraditionalForm], 159 | "errors" -> {}, "input" -> "a=0", "state" -> "Boxes"|>, 160 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 161 | "b80fa055-a0e5-4ec8-b294-1fd7b60023d3"], 162 | " is given in (3)." 163 | }], "Text", 164 | CellChangeTimes->{{3.9493964472838182`*^9, 3.949396473776319*^9}, { 165 | 3.949396520439193*^9, 3.9493965449364357`*^9}, {3.949425759058935*^9, 166 | 3.949425787128389*^9}},ExpressionUUID->"b8df313e-6ca1-4a83-bebf-\ 167 | 39b1523ce039"], 168 | 169 | Cell[BoxData[{ 170 | RowBox[{ 171 | RowBox[{"f", 172 | RowBox[{"(", "0", ")"}]}], "=", 173 | RowBox[{ 174 | SubscriptBox["lim", 175 | RowBox[{"n", "->", "\[Infinity]"}]], 176 | RowBox[{ 177 | UnderoverscriptBox["\[Sum]", 178 | RowBox[{"j", "=", "0"}], "n"], 179 | RowBox[{ 180 | FractionBox[ 181 | RowBox[{ 182 | SuperscriptBox["f", 183 | RowBox[{"(", "j", ")"}]], 184 | RowBox[{"(", "0", ")"}]}], 185 | RowBox[{"j", "!"}]], 186 | SuperscriptBox["\[Theta]", "j"]}]}]}]}], "\[IndentingNewLine]", 187 | RowBox[{ 188 | RowBox[{"f", 189 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 190 | RowBox[{ 191 | RowBox[{ 192 | FractionBox[ 193 | RowBox[{"f", 194 | RowBox[{"(", "0", ")"}]}], 195 | RowBox[{"0", "!"}]], 196 | SuperscriptBox["\[Theta]", "0"]}], "+", 197 | RowBox[{ 198 | FractionBox[ 199 | RowBox[{ 200 | RowBox[{"f", "'"}], 201 | RowBox[{"(", "0", ")"}]}], 202 | RowBox[{"1", "!"}]], 203 | SuperscriptBox["\[Theta]", "1"]}], "+", 204 | RowBox[{ 205 | FractionBox[ 206 | RowBox[{ 207 | RowBox[{"f", "''"}], 208 | RowBox[{"(", "0", ")"}]}], 209 | RowBox[{"2", "!"}]], 210 | SuperscriptBox["\[Theta]", "2"]}], "+", 211 | RowBox[{ 212 | FractionBox[ 213 | RowBox[{ 214 | SuperscriptBox["f", 215 | RowBox[{"(", "3", ")"}]], 216 | RowBox[{"(", "0", ")"}]}], 217 | RowBox[{"3", "!"}]], 218 | SuperscriptBox["\[Theta]", "3"]}], "+", 219 | "\[Ellipsis]"}]}]}], "DisplayFormulaNumbered", 220 | CellChangeTimes->{{3.9469112784188538`*^9, 3.946911359464532*^9}, { 221 | 3.946911390054572*^9, 3.9469114301328583`*^9}, {3.946911488918517*^9, 222 | 3.9469115622648907`*^9}, {3.949399218084553*^9, 3.949399230662916*^9}, { 223 | 3.94942559825988*^9, 224 | 3.949425599610846*^9}},ExpressionUUID->"48b3c032-438b-46a9-bf84-\ 225 | ccdd7e316a6c"], 226 | 227 | Cell[TextData[{ 228 | "As an example of such an expansion, we consider ", 229 | Cell[BoxData[ 230 | FormBox[ 231 | RowBox[{ 232 | RowBox[{"f", "(", "\[Theta]", ")"}], "=", 233 | RowBox[{"cos", "(", "\[Theta]", ")"}]}], TraditionalForm]],ExpressionUUID-> 234 | "d3af7252-1319-4553-91f2-361dd28c146e"], 235 | ". In (4), we calculate the first six derivatives of ", 236 | Cell[BoxData[ 237 | FormBox[ 238 | RowBox[{"cos", "(", "\[Theta]", ")"}], TraditionalForm]],ExpressionUUID-> 239 | "ea007d77-23d8-4d8a-ab73-0d3dbe56c423"], 240 | "." 241 | }], "Text", 242 | CellChangeTimes->{{3.946911734020653*^9, 3.946911780632935*^9}, 243 | 3.946911829341666*^9, {3.94691705809733*^9, 244 | 3.946917058160486*^9}},ExpressionUUID->"fa6cafa4-d884-4878-bdff-\ 245 | 1b175d8afaf9"], 246 | 247 | Cell[BoxData[{ 248 | RowBox[{ 249 | RowBox[{"f", 250 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 251 | RowBox[{"cos", 252 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 253 | RowBox[{ 254 | RowBox[{ 255 | RowBox[{"f", "'"}], 256 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 257 | RowBox[{ 258 | RowBox[{"-", "sin"}], 259 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 260 | RowBox[{ 261 | RowBox[{ 262 | RowBox[{"f", "''"}], 263 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 264 | RowBox[{ 265 | RowBox[{"-", "cos"}], 266 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 267 | RowBox[{ 268 | RowBox[{ 269 | SuperscriptBox["f", 270 | RowBox[{"(", "3", ")"}]], 271 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 272 | RowBox[{"sin", 273 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 274 | RowBox[{ 275 | RowBox[{ 276 | SuperscriptBox["f", 277 | RowBox[{"(", "4", ")"}]], 278 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 279 | RowBox[{"cos", 280 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 281 | RowBox[{ 282 | RowBox[{ 283 | SuperscriptBox["f", 284 | RowBox[{"(", "5", ")"}]], 285 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 286 | RowBox[{ 287 | RowBox[{"-", "sin"}], 288 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 289 | RowBox[{ 290 | RowBox[{ 291 | SuperscriptBox["f", 292 | RowBox[{"(", "6", ")"}]], 293 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 294 | RowBox[{ 295 | RowBox[{"-", "cos"}], 296 | RowBox[{ 297 | "(", "\[Theta]", 298 | ")"}]}]}], "\[IndentingNewLine]", "\[VerticalEllipsis]"}], \ 299 | "DisplayFormulaNumbered", 300 | CellChangeTimes->{{3.946911581833333*^9, 3.946911651583643*^9}, { 301 | 3.946911784445849*^9, 3.946911798395643*^9}, {3.946917009856955*^9, 302 | 3.946917051998782*^9}},ExpressionUUID->"222b91e2-00ff-4d0f-9b91-\ 303 | 752984836119"], 304 | 305 | Cell[TextData[{ 306 | "The ", 307 | StyleBox["Table", "CodeText"], 308 | " function can be used to generate multiple calculations. We use it below to \ 309 | calculate the first six derivative of the cosine function with respect to ", 310 | Cell[BoxData[ 311 | FormBox["\[Theta]", TraditionalForm]],ExpressionUUID-> 312 | "25f7a25a-f730-40a9-95aa-a52983f03080"], 313 | "." 314 | }], "Text", 315 | CellChangeTimes->{{3.946912109045138*^9, 3.946912150915388*^9}, 316 | 3.9469127174198437`*^9, {3.9469130492940702`*^9, 3.9469130514313803`*^9}, { 317 | 3.946913161165059*^9, 318 | 3.946913161243032*^9}},ExpressionUUID->"c5fe48d0-04f3-4c6e-9900-\ 319 | 79eb15c4ceb5"], 320 | 321 | Cell[CellGroupData[{ 322 | 323 | Cell[BoxData[ 324 | RowBox[{ 325 | RowBox[{"(*", 326 | RowBox[{ 327 | "Calculate", " ", "the", " ", "first", " ", "six", " ", "derivatives", " ", 328 | "of", " ", "the", " ", "cosine", " ", "function", " ", "with", " ", 329 | "respect", " ", "to", " ", Cell[ 330 | "\[Theta]",ExpressionUUID->"1c5f10f1-9d1a-46e5-bae6-12666f3e4745"]}], 331 | "*)"}], "\[IndentingNewLine]", 332 | RowBox[{"Table", "[", 333 | RowBox[{ 334 | RowBox[{"D", "[", 335 | RowBox[{ 336 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 337 | RowBox[{"{", 338 | RowBox[{"\[Theta]", ",", "j"}], "}"}]}], "]"}], ",", 339 | RowBox[{"{", 340 | RowBox[{"j", ",", "1", ",", "6"}], "}"}]}], "]"}]}]], "Input", 341 | CellChangeTimes->{{3.9469118625722637`*^9, 3.9469118865659*^9}, { 342 | 3.946911919282502*^9, 3.946911926829547*^9}, {3.946912079378166*^9, 343 | 3.946912097366438*^9}, {3.946912154619238*^9, 3.946912188640149*^9}, { 344 | 3.946913042804163*^9, 3.9469130550696583`*^9}, {3.9469131544272203`*^9, 345 | 3.9469131545387087`*^9}, {3.946917083083516*^9, 3.9469170831021*^9}}, 346 | CellLabel->"In[1]:=",ExpressionUUID->"c4f6590d-e7d5-4414-938d-30b83f0c4233"], 347 | 348 | Cell[BoxData[ 349 | RowBox[{"{", 350 | RowBox[{ 351 | RowBox[{"-", 352 | RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", 353 | RowBox[{"-", 354 | RowBox[{"Cos", "[", "\[Theta]", "]"}]}], ",", 355 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 356 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 357 | RowBox[{"-", 358 | RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", 359 | RowBox[{"-", 360 | RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], "}"}]], "Output", 361 | CellChangeTimes->{3.9499285424039307`*^9}, 362 | CellLabel->"Out[1]=",ExpressionUUID->"271552bc-b193-4c0b-a0f6-d55a22469680"] 363 | }, Open ]], 364 | 365 | Cell[TextData[{ 366 | "In (5), we calculate the derivates at ", 367 | Cell[BoxData[ 368 | FormBox[ 369 | TemplateBox[<|"boxes" -> FormBox[ 370 | RowBox[{"\[Theta]", "\[LongEqual]", "0"}], TraditionalForm], 371 | "errors" -> {}, "input" -> "\\theta = 0", "state" -> "Boxes"|>, 372 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 373 | "16b377a2-80fb-4be6-b408-1ae4fb9202f7"], 374 | "." 375 | }], "Text", 376 | CellChangeTimes->{{3.94691182068176*^9, 377 | 3.946911852211523*^9}},ExpressionUUID->"4bb1173f-116c-4e12-b8a7-\ 378 | 15950abc9b0f"], 379 | 380 | Cell[BoxData[{ 381 | RowBox[{ 382 | RowBox[{"f", 383 | RowBox[{"(", "0", ")"}]}], "=", 384 | RowBox[{ 385 | RowBox[{"cos", 386 | RowBox[{"(", "0", ")"}]}], "=", "1"}]}], "\[IndentingNewLine]", 387 | RowBox[{ 388 | RowBox[{ 389 | RowBox[{"f", "'"}], 390 | RowBox[{"(", "0", ")"}]}], "=", 391 | RowBox[{ 392 | RowBox[{ 393 | RowBox[{"-", "sin"}], 394 | RowBox[{"(", "0", ")"}]}], "=", "0"}]}], "\[IndentingNewLine]", 395 | RowBox[{ 396 | RowBox[{ 397 | RowBox[{"f", "''"}], 398 | RowBox[{"(", "0", ")"}]}], "=", 399 | RowBox[{ 400 | RowBox[{ 401 | RowBox[{"-", "cos"}], 402 | RowBox[{"(", "0", ")"}]}], "=", 403 | RowBox[{"-", "1"}]}]}], "\[IndentingNewLine]", 404 | RowBox[{ 405 | RowBox[{ 406 | SuperscriptBox["f", 407 | RowBox[{"(", "3", ")"}]], 408 | RowBox[{"(", "0", ")"}]}], "=", 409 | RowBox[{ 410 | RowBox[{"sin", 411 | RowBox[{"(", "0", ")"}]}], "=", "0"}]}], "\[IndentingNewLine]", 412 | RowBox[{ 413 | RowBox[{ 414 | SuperscriptBox["f", 415 | RowBox[{"(", "4", ")"}]], 416 | RowBox[{"(", "0", ")"}]}], "=", 417 | RowBox[{ 418 | RowBox[{"cos", 419 | RowBox[{"(", "0", ")"}]}], "=", "1"}]}], "\[IndentingNewLine]", 420 | RowBox[{ 421 | RowBox[{ 422 | SuperscriptBox["f", 423 | RowBox[{"(", "5", ")"}]], 424 | RowBox[{"(", "0", ")"}]}], "=", 425 | RowBox[{ 426 | RowBox[{ 427 | RowBox[{"-", "sin"}], 428 | RowBox[{"(", "0", ")"}]}], "=", "0"}]}], "\[IndentingNewLine]", 429 | RowBox[{ 430 | RowBox[{ 431 | SuperscriptBox["f", 432 | RowBox[{"(", "6", ")"}]], 433 | RowBox[{"(", "0", ")"}]}], "=", 434 | RowBox[{ 435 | RowBox[{ 436 | RowBox[{"-", "cos"}], 437 | RowBox[{"(", "0", ")"}]}], "=", 438 | RowBox[{ 439 | "-", "1"}]}]}], "\[IndentingNewLine]", "\[VerticalEllipsis]"}], \ 440 | "DisplayFormulaNumbered", 441 | CellChangeTimes->{{3.946911665005094*^9, 3.9469117015528917`*^9}, { 442 | 3.946911801546775*^9, 3.946911815617683*^9}, {3.946917105639243*^9, 443 | 3.9469171284645844`*^9}},ExpressionUUID->"c1f4d054-59e7-4be1-a69a-\ 444 | 8ad96b666a54"], 445 | 446 | Cell[TextData[{ 447 | "We can also verify the results at ", 448 | Cell[BoxData[ 449 | FormBox[ 450 | RowBox[{"\[Theta]", "=", "0"}], TraditionalForm]],ExpressionUUID-> 451 | "411a83eb-8769-472d-bf1e-302bda63f7a0"], 452 | " using the ", 453 | StyleBox["ReplaceAll", "CodeText"], 454 | " function. We do so for the first seven derivatives." 455 | }], "Text", 456 | CellChangeTimes->{{3.94691226864566*^9, 3.946912294185821*^9}, { 457 | 3.9469130320937023`*^9, 3.94691303858547*^9}, {3.946913147654008*^9, 458 | 3.946913147777444*^9}, {3.949396827269679*^9, 459 | 3.9493968280153522`*^9}},ExpressionUUID->"e757bd7a-9b65-4c7a-a3cc-\ 460 | d9e0b0dd6a42"], 461 | 462 | Cell[CellGroupData[{ 463 | 464 | Cell[BoxData[ 465 | RowBox[{ 466 | RowBox[{"(*", 467 | RowBox[{ 468 | "Calculate", " ", "the", " ", "value", " ", "of", " ", "the", " ", "first", 469 | " ", "seven", " ", "derivatives", " ", "at", " ", Cell[ 470 | "\[Theta]=0",ExpressionUUID->"d3efcb08-33da-41a9-97ce-f4dff0ed9c07"]}], 471 | "*)"}], "\[IndentingNewLine]", 472 | RowBox[{"ReplaceAll", "[", 473 | RowBox[{ 474 | RowBox[{"Table", "[", 475 | RowBox[{ 476 | RowBox[{"D", "[", 477 | RowBox[{ 478 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 479 | RowBox[{"{", 480 | RowBox[{"\[Theta]", ",", "j"}], "}"}]}], "]"}], ",", 481 | RowBox[{"{", 482 | RowBox[{"j", ",", "0", ",", "6"}], "}"}]}], "]"}], ",", 483 | RowBox[{"\[Theta]", "->", "0"}]}], "]"}]}]], "Input", 484 | CellChangeTimes->{{3.946912341435441*^9, 3.9469123884737577`*^9}, { 485 | 3.9469127814114943`*^9, 3.946912808478808*^9}, {3.946913017670539*^9, 486 | 3.9469130278615417`*^9}, {3.9469131349080276`*^9, 3.946913142972316*^9}, { 487 | 3.949396816410626*^9, 3.949396832428729*^9}}, 488 | CellLabel->"In[2]:=",ExpressionUUID->"7a1301b6-6e8d-4e04-96e0-948cdedef374"], 489 | 490 | Cell[BoxData[ 491 | RowBox[{"{", 492 | RowBox[{"1", ",", "0", ",", 493 | RowBox[{"-", "1"}], ",", "0", ",", "1", ",", "0", ",", 494 | RowBox[{"-", "1"}]}], "}"}]], "Output", 495 | CellChangeTimes->{3.949928576385254*^9}, 496 | CellLabel->"Out[2]=",ExpressionUUID->"6c28df34-0261-44fe-a6c6-fa887da9fe0c"] 497 | }, Open ]], 498 | 499 | Cell[TextData[{ 500 | "In (6) we substitute all the results, starting with the fact that the \ 501 | cosine of ", 502 | Cell[BoxData[ 503 | FormBox["0", TraditionalForm]],ExpressionUUID-> 504 | "9b8ca33b-cc34-489a-9ee3-6d40f9df7e4e"], 505 | " is ", 506 | Cell[BoxData[ 507 | FormBox["1", TraditionalForm]],ExpressionUUID-> 508 | "ac39784d-0be7-4cdb-b10e-83eb4cb0f083"], 509 | ". " 510 | }], "Text", 511 | CellChangeTimes->{{3.9469127456920843`*^9, 3.946912758168522*^9}, { 512 | 3.94691284850578*^9, 513 | 3.94691286841567*^9}},ExpressionUUID->"a446de9c-651a-49da-b3cb-\ 514 | 052d4acd188e"], 515 | 516 | Cell[BoxData[{ 517 | RowBox[{ 518 | RowBox[{"f", 519 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 520 | RowBox[{"cos", 521 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 522 | RowBox[{ 523 | RowBox[{"cos", 524 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 525 | RowBox[{ 526 | RowBox[{ 527 | FractionBox[ 528 | RowBox[{"f", 529 | RowBox[{"(", "0", ")"}]}], 530 | RowBox[{"0", "!"}]], 531 | SuperscriptBox["\[Theta]", "0"]}], "+", 532 | RowBox[{ 533 | FractionBox[ 534 | RowBox[{ 535 | RowBox[{"f", "'"}], 536 | RowBox[{"(", "0", ")"}]}], 537 | RowBox[{"1", "!"}]], 538 | SuperscriptBox["\[Theta]", "1"]}], "+", 539 | RowBox[{ 540 | FractionBox[ 541 | RowBox[{ 542 | RowBox[{"f", "''"}], 543 | RowBox[{"(", "0", ")"}]}], 544 | RowBox[{"2", "!"}]], 545 | SuperscriptBox["\[Theta]", "2"]}], "+", 546 | RowBox[{ 547 | FractionBox[ 548 | RowBox[{ 549 | SuperscriptBox["f", 550 | RowBox[{"(", "3", ")"}]], 551 | RowBox[{"(", "0", ")"}]}], 552 | RowBox[{"3", "!"}]], 553 | SuperscriptBox["\[Theta]", "3"]}], "+", "\[Ellipsis]"}]}], "\[IndentingNewLine]", 554 | RowBox[{ 555 | RowBox[{"cos", 556 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 557 | RowBox[{ 558 | RowBox[{ 559 | FractionBox["1", 560 | RowBox[{"0", "!"}]], 561 | SuperscriptBox["\[Theta]", "0"]}], "+", 562 | RowBox[{ 563 | FractionBox["0", 564 | RowBox[{"1", "!"}]], "\[Theta]"}], "+", 565 | RowBox[{ 566 | FractionBox[ 567 | RowBox[{"-", "1"}], 568 | RowBox[{"2", "!"}]], 569 | SuperscriptBox["\[Theta]", "2"]}], "+", 570 | RowBox[{ 571 | FractionBox["0", 572 | RowBox[{"3", "!"}]], 573 | SuperscriptBox["\[Theta]", "3"]}], "+", 574 | RowBox[{ 575 | FractionBox["1", 576 | RowBox[{"4", "!"}]], 577 | SuperscriptBox["\[Theta]", "4"]}], "+", 578 | RowBox[{ 579 | FractionBox["0", 580 | RowBox[{"5", "!"}]], 581 | SuperscriptBox["\[Theta]", "5"]}], "+", 582 | RowBox[{ 583 | FractionBox[ 584 | RowBox[{"-", "1"}], 585 | RowBox[{"6", "!"}]], 586 | SuperscriptBox["\[Theta]", "6"]}], "+", "\[Ellipsis]"}]}], "\[IndentingNewLine]", 587 | RowBox[{ 588 | RowBox[{"cos", 589 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 590 | RowBox[{"1", "-", 591 | FractionBox[ 592 | SuperscriptBox["\[Theta]", "2"], 593 | RowBox[{"2", "!"}]], "+", 594 | FractionBox[ 595 | SuperscriptBox["\[Theta]", "4"], 596 | RowBox[{"4", "!"}]], "-", 597 | FractionBox[ 598 | SuperscriptBox["\[Theta]", "6"], 599 | RowBox[{"6", "!"}]], "+", "\[Ellipsis]"}]}]}], "DisplayFormulaNumbered", 600 | CellChangeTimes->{{3.946912398753281*^9, 3.946912401835047*^9}, { 601 | 3.946912489011959*^9, 3.946912604629195*^9}, {3.9469128783657503`*^9, 602 | 3.9469130013477573`*^9}, {3.9469130825259323`*^9, 3.946913127804245*^9}, { 603 | 3.949396664283824*^9, 604 | 3.94939667110186*^9}},ExpressionUUID->"871678df-4a10-4c39-ad62-\ 605 | 8d744e85e6f6"], 606 | 607 | Cell[TextData[{ 608 | "It is simpler to use the ", 609 | StyleBox["Manipulate", "CodeText"], 610 | " function to demonstrate the Maclaurin series of the cosine function. Note \ 611 | that the denominators show the factorial values." 612 | }], "Text", 613 | CellChangeTimes->{{3.946913195837162*^9, 3.946913231813004*^9}, { 614 | 3.946913448348699*^9, 3.9469134548268843`*^9}, {3.946917182982242*^9, 615 | 3.946917206292921*^9}},ExpressionUUID->"79890846-eb15-442b-8421-\ 616 | b6d79c93afb4"], 617 | 618 | Cell[CellGroupData[{ 619 | 620 | Cell[BoxData[ 621 | RowBox[{ 622 | RowBox[{"(*", 623 | RowBox[{ 624 | "Approximate", " ", "the", " ", "cosine", " ", "function", " ", "around", " ", 625 | Cell["\[Theta]=0",ExpressionUUID-> 626 | "9e3d08c2-6005-4964-92a5-5e66e5c0862b"], " ", "with", " ", "more", " ", 627 | "and", " ", "more", " ", "terms"}], "*)"}], "\[IndentingNewLine]", 628 | RowBox[{"Manipulate", "[", 629 | RowBox[{ 630 | RowBox[{"Normal", "[", 631 | RowBox[{"Evaluate", "[", 632 | RowBox[{"Series", "[", 633 | RowBox[{ 634 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 635 | RowBox[{"{", 636 | RowBox[{"\[Theta]", ",", "0", ",", "j"}], "}"}]}], "]"}], "]"}], 637 | "]"}], ",", 638 | RowBox[{"{", 639 | RowBox[{"j", ",", "2", ",", "20", ",", "2"}], "}"}]}], "]"}]}]], "Input",\ 640 | 641 | CellChangeTimes->{{3.9469079427166233`*^9, 3.9469079665158453`*^9}, { 642 | 3.9469085045521183`*^9, 3.946908508487586*^9}, {3.946913262910437*^9, 643 | 3.946913264484158*^9}, {3.946913437681933*^9, 3.9469134408085117`*^9}, { 644 | 3.9469135284830914`*^9, 3.94691354984505*^9}}, 645 | CellLabel->"In[3]:=",ExpressionUUID->"c05b281d-b8bc-42a5-b677-5c18c40bfa78"], 646 | 647 | Cell[BoxData[ 648 | TagBox[ 649 | StyleBox[ 650 | DynamicModuleBox[{$CellContext`j$$ = 20, Typeset`show$$ = True, 651 | Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 652 | Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 653 | "\"untitled\"", Typeset`specs$$ = {{ 654 | Hold[$CellContext`j$$], 2, 20, 2, ControlType -> Manipulator}}, 655 | Typeset`size$$ = { 656 | 1400.2079472656249`, {14.4543598046875, 29.283405195312504`}}, 657 | Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, 658 | Typeset`keyframeActionsQ$$ = False, Typeset`keyframeList$$ = {}}, 659 | DynamicBox[Manipulate`ManipulateBoxes[ 660 | 1, StandardForm, "Variables" :> {$CellContext`j$$ = 2}, 661 | "ControllerVariables" :> {}, 662 | "OtherVariables" :> { 663 | Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 664 | Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 665 | Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, 666 | Typeset`skipInitDone$$, Typeset`keyframeActionsQ$$, 667 | Typeset`keyframeList$$}, "Body" :> Normal[ 668 | Evaluate[ 669 | Series[ 670 | 671 | Cos[$CellContext`\[Theta]], {$CellContext`\[Theta], 672 | 0, $CellContext`j$$}]]], 673 | "Specifications" :> {{$CellContext`j$$, 2, 20, 2}}, "Options" :> {}, 674 | "DefaultOptions" :> {}], 675 | ImageSizeCache->{ 676 | 1469.857947265625, {87.07557042724609, 95.67347081787109}}, 677 | SingleEvaluation->True], 678 | Deinitialization:>None, 679 | DynamicModuleValues:>{}, 680 | SynchronousInitialization->True, 681 | UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, 682 | UnsavedVariables:>{Typeset`initDone$$}, 683 | UntrackedVariables:>{Typeset`size$$}], "Manipulate", 684 | Deployed->True, 685 | StripOnInput->False], 686 | Manipulate`InterpretManipulate[1]]], "Output", 687 | CellChangeTimes->{3.949928665696015*^9}, 688 | CellLabel->"Out[3]=",ExpressionUUID->"cc2d7473-fd1f-46d5-bbb7-85ef1af0af8a"] 689 | }, Open ]], 690 | 691 | Cell["\<\ 692 | The visual interpretation of the Maclaurin series using more and more terms \ 693 | is very striking. the more terms we use, the better the approximation of the \ 694 | cosine function.\ 695 | \>", "Text", 696 | CellChangeTimes->{{3.9469132899976673`*^9, 3.946913313476039*^9}, { 697 | 3.9469134584170856`*^9, 698 | 3.946913473908882*^9}},ExpressionUUID->"9bf18f1a-3aa8-4ff5-95a6-\ 699 | e28ccd715941"], 700 | 701 | Cell[CellGroupData[{ 702 | 703 | Cell[BoxData[ 704 | RowBox[{ 705 | RowBox[{"(*", 706 | RowBox[{ 707 | "Approximate", " ", "the", " ", "cosine", " ", "function", " ", "around", " ", 708 | Cell["\[Theta]=0",ExpressionUUID-> 709 | "5966e10b-fc60-4cf0-bd5a-edd2d61210e4"], " ", "with", " ", "an", " ", 710 | "increasing", " ", "number", " ", "of", " ", "terms"}], "*)"}], 711 | "\[IndentingNewLine]", 712 | RowBox[{"Manipulate", "[", 713 | RowBox[{ 714 | RowBox[{"Column", "[", 715 | RowBox[{"{", 716 | RowBox[{ 717 | RowBox[{"Normal", "[", 718 | RowBox[{"Evaluate", "[", 719 | RowBox[{"Series", "[", 720 | RowBox[{ 721 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 722 | RowBox[{"{", 723 | RowBox[{"\[Theta]", ",", "0", ",", "j"}], "}"}]}], "]"}], "]"}], 724 | "]"}], ",", 725 | RowBox[{"Plot", "[", 726 | RowBox[{ 727 | RowBox[{"{", 728 | RowBox[{ 729 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 730 | RowBox[{"Evaluate", "[", 731 | RowBox[{"Normal", "[", 732 | RowBox[{"Series", "[", 733 | RowBox[{ 734 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 735 | RowBox[{"{", 736 | RowBox[{"\[Theta]", ",", "0", ",", "j"}], "}"}]}], "]"}], 737 | "]"}], "]"}]}], "}"}], ",", 738 | RowBox[{"{", 739 | RowBox[{"\[Theta]", ",", 740 | RowBox[{ 741 | RowBox[{"-", "5"}], " ", "\[Pi]"}], ",", 742 | RowBox[{"5", " ", "\[Pi]"}]}], "}"}], ",", 743 | RowBox[{"PlotRange", "->", 744 | RowBox[{"{", 745 | RowBox[{ 746 | RowBox[{"{", 747 | RowBox[{ 748 | RowBox[{ 749 | RowBox[{"-", "5"}], " ", "\[Pi]"}], ",", 750 | RowBox[{"5", " ", "\[Pi]"}]}], "}"}], ",", 751 | RowBox[{"{", 752 | RowBox[{ 753 | RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}], ",", 754 | RowBox[{"GridLines", "->", "Automatic"}], ",", 755 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]}], "}"}], "]"}], ",", 756 | RowBox[{"{", 757 | RowBox[{"j", ",", "2", ",", "24", ",", "2"}], "}"}]}], "]"}]}]], "Input",\ 758 | 759 | CellChangeTimes->{{3.946908198186825*^9, 3.946908403468902*^9}, { 760 | 3.946908434883709*^9, 3.946908484116255*^9}, {3.946908630784532*^9, 761 | 3.946908665756588*^9}, {3.946913335304104*^9, 3.94691334970324*^9}, { 762 | 3.946913490086856*^9, 3.9469135196524057`*^9}, {3.949396928232163*^9, 763 | 3.9493969900145607`*^9}, {3.949397050013983*^9, 3.949397052008428*^9}}, 764 | CellLabel->"In[4]:=",ExpressionUUID->"cbd78080-dd56-4eed-917b-0913712b83eb"], 765 | 766 | Cell[BoxData[ 767 | TagBox[ 768 | StyleBox[ 769 | DynamicModuleBox[{$CellContext`j$$ = 24, Typeset`show$$ = True, 770 | Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 771 | Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 772 | "\"untitled\"", Typeset`specs$$ = {{ 773 | Hold[$CellContext`j$$], 2, 24, 2, ControlType -> Manipulator}}, 774 | Typeset`size$$ = { 775 | 1555.1172556640622`, {290.94023570870473`, 299.53813609932973`}}, 776 | Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, 777 | Typeset`keyframeActionsQ$$ = False, Typeset`keyframeList$$ = {}}, 778 | DynamicBox[Manipulate`ManipulateBoxes[ 779 | 1, StandardForm, "Variables" :> {$CellContext`j$$ = 2}, 780 | "ControllerVariables" :> {}, 781 | "OtherVariables" :> { 782 | Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 783 | Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 784 | Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, 785 | Typeset`skipInitDone$$, Typeset`keyframeActionsQ$$, 786 | Typeset`keyframeList$$}, "Body" :> Column[{ 787 | Normal[ 788 | Evaluate[ 789 | Series[ 790 | 791 | Cos[$CellContext`\[Theta]], {$CellContext`\[Theta], 792 | 0, $CellContext`j$$}]]], 793 | Plot[{ 794 | Cos[$CellContext`\[Theta]], 795 | Evaluate[ 796 | Normal[ 797 | Series[ 798 | 799 | Cos[$CellContext`\[Theta]], {$CellContext`\[Theta], 800 | 0, $CellContext`j$$}]]]}, {$CellContext`\[Theta], (-5) Pi, 5 801 | Pi}, PlotRange -> {{(-5) Pi, 5 Pi}, {-2, 2}}, GridLines -> 802 | Automatic, ImageSize -> Large]}], 803 | "Specifications" :> {{$CellContext`j$$, 2, 24, 2}}, "Options" :> {}, 804 | "DefaultOptions" :> {}], 805 | ImageSizeCache->{ 806 | 1624.7672556640623`, {360.4458738312633, 369.0437742218883}}, 807 | SingleEvaluation->True], 808 | Deinitialization:>None, 809 | DynamicModuleValues:>{}, 810 | SynchronousInitialization->True, 811 | UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, 812 | UnsavedVariables:>{Typeset`initDone$$}, 813 | UntrackedVariables:>{Typeset`size$$}], "Manipulate", 814 | Deployed->True, 815 | StripOnInput->False], 816 | Manipulate`InterpretManipulate[1]]], "Output", 817 | CellChangeTimes->{3.949928693443075*^9}, 818 | CellLabel->"Out[4]=",ExpressionUUID->"c83b7dd9-812d-4ed1-975d-0a437e0def88"] 819 | }, Open ]], 820 | 821 | Cell["\<\ 822 | The same can be done with the sine function. We see the function and its \ 823 | first four derivatives in (7).\ 824 | \>", "Text", 825 | CellChangeTimes->{{3.946913361290986*^9, 3.94691339730093*^9}, { 826 | 3.949397576118857*^9, 827 | 3.9493975989436827`*^9}},ExpressionUUID->"982fbac2-0444-4806-b90c-\ 828 | 0147f3e125c1"], 829 | 830 | Cell[BoxData[{ 831 | RowBox[{ 832 | RowBox[{"f", 833 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 834 | RowBox[{"sin", 835 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 836 | RowBox[{ 837 | RowBox[{ 838 | RowBox[{"f", "'"}], 839 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 840 | RowBox[{"cos", 841 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 842 | RowBox[{ 843 | RowBox[{ 844 | RowBox[{"f", "''"}], 845 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 846 | RowBox[{ 847 | RowBox[{"-", "sin"}], 848 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 849 | RowBox[{ 850 | RowBox[{ 851 | SuperscriptBox["f", 852 | RowBox[{"(", "3", ")"}]], 853 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 854 | RowBox[{ 855 | RowBox[{"-", "cos"}], 856 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 857 | RowBox[{ 858 | RowBox[{ 859 | SuperscriptBox["f", 860 | RowBox[{"(", "4", ")"}]], 861 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 862 | RowBox[{"sin", 863 | RowBox[{ 864 | "(", "\[Theta]", 865 | ")"}]}]}], "\[IndentingNewLine]", "\[VerticalEllipsis]"}], \ 866 | "DisplayFormulaNumbered", 867 | CellChangeTimes->{{3.949397522045751*^9, 868 | 3.949397569343399*^9}},ExpressionUUID->"b088b2d7-e9bc-4b53-a749-\ 869 | e7b3c93a3dda"], 870 | 871 | Cell[TextData[{ 872 | "In (8), we substitute ", 873 | Cell[BoxData[ 874 | FormBox[ 875 | RowBox[{"\[Theta]", "=", "0"}], TraditionalForm]],ExpressionUUID-> 876 | "b09423f2-766b-43d1-9f0f-cf48b300ea1f"], 877 | "." 878 | }], "Text", 879 | CellChangeTimes->{{3.949397603570773*^9, 880 | 3.949397613335608*^9}},ExpressionUUID->"74622326-bf7a-47c4-8658-\ 881 | 85b091169094"], 882 | 883 | Cell[BoxData[{ 884 | RowBox[{ 885 | RowBox[{"f", 886 | RowBox[{"(", "0", ")"}]}], "=", 887 | RowBox[{ 888 | RowBox[{"sin", 889 | RowBox[{"(", "0", ")"}]}], "=", "0"}]}], "\[IndentingNewLine]", 890 | RowBox[{ 891 | RowBox[{ 892 | RowBox[{"f", "'"}], 893 | RowBox[{"(", "0", ")"}]}], "=", 894 | RowBox[{ 895 | RowBox[{"cos", 896 | RowBox[{"(", "0", ")"}]}], "=", "1"}]}], "\[IndentingNewLine]", 897 | RowBox[{ 898 | RowBox[{ 899 | RowBox[{"f", "''"}], 900 | RowBox[{"(", "0", ")"}]}], "=", 901 | RowBox[{ 902 | RowBox[{ 903 | RowBox[{"-", "sin"}], 904 | RowBox[{"(", "0", ")"}]}], "=", "0"}]}], "\[IndentingNewLine]", 905 | RowBox[{ 906 | RowBox[{ 907 | SuperscriptBox["f", 908 | RowBox[{"(", "3", ")"}]], 909 | RowBox[{"(", "0", ")"}]}], "=", 910 | RowBox[{ 911 | RowBox[{ 912 | RowBox[{"-", "cos"}], 913 | RowBox[{"(", "0", ")"}]}], "=", 914 | RowBox[{"-", "1"}]}]}], "\[IndentingNewLine]", 915 | RowBox[{ 916 | RowBox[{ 917 | SuperscriptBox["f", 918 | RowBox[{"(", "4", ")"}]], 919 | RowBox[{"(", "0", ")"}]}], "=", 920 | RowBox[{ 921 | RowBox[{"sin", 922 | RowBox[{"(", "0", ")"}]}], "=", 923 | "0"}]}], "\[IndentingNewLine]", "\[VerticalEllipsis]"}], \ 924 | "DisplayFormulaNumbered", 925 | CellChangeTimes->{{3.9493976281718197`*^9, 926 | 3.949397659826126*^9}},ExpressionUUID->"0b4d0c52-cbf2-4687-86db-\ 927 | 4b43696ac384"], 928 | 929 | Cell[TextData[{ 930 | "We can also verify the results at ", 931 | Cell[BoxData[ 932 | FormBox[ 933 | RowBox[{"\[Theta]", "=", "0"}], TraditionalForm]],ExpressionUUID-> 934 | "bee1feb3-76d9-4147-83a3-d94c8294d047"], 935 | " using the ", 936 | StyleBox["ReplaceAll", "CodeText"], 937 | " function. We do so for the first seven derivatives." 938 | }], "Text", 939 | CellChangeTimes->{{3.94691226864566*^9, 3.946912294185821*^9}, { 940 | 3.9469130320937023`*^9, 3.94691303858547*^9}, {3.946913147654008*^9, 941 | 3.946913147777444*^9}, {3.949396827269679*^9, 942 | 3.9493968280153522`*^9}},ExpressionUUID->"b866f5f3-8d2d-4ac8-b596-\ 943 | 2e5d871ba899"], 944 | 945 | Cell[CellGroupData[{ 946 | 947 | Cell[BoxData[ 948 | RowBox[{ 949 | RowBox[{"(*", 950 | RowBox[{ 951 | "Calculate", " ", "the", " ", "value", " ", "of", " ", "the", " ", "first", 952 | " ", "seven", " ", "derivatives", " ", "at", " ", Cell[ 953 | "\[Theta]=0",ExpressionUUID->"0c796451-edeb-446b-83d1-b100880462fe"]}], 954 | "*)"}], "\[IndentingNewLine]", 955 | RowBox[{"ReplaceAll", "[", 956 | RowBox[{ 957 | RowBox[{"Table", "[", 958 | RowBox[{ 959 | RowBox[{"D", "[", 960 | RowBox[{ 961 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 962 | RowBox[{"{", 963 | RowBox[{"\[Theta]", ",", "j"}], "}"}]}], "]"}], ",", 964 | RowBox[{"{", 965 | RowBox[{"j", ",", "0", ",", "6"}], "}"}]}], "]"}], ",", 966 | RowBox[{"\[Theta]", "->", "0"}]}], "]"}]}]], "Input", 967 | CellChangeTimes->{{3.946912341435441*^9, 3.9469123884737577`*^9}, { 968 | 3.9469127814114943`*^9, 3.946912808478808*^9}, {3.946913017670539*^9, 969 | 3.9469130278615417`*^9}, {3.9469131349080276`*^9, 3.946913142972316*^9}, { 970 | 3.949396816410626*^9, 3.949396832428729*^9}, {3.949397696773507*^9, 971 | 3.949397697557963*^9}}, 972 | CellLabel->"In[5]:=",ExpressionUUID->"324fcb3a-62cd-40ea-8b51-a5661155b587"], 973 | 974 | Cell[BoxData[ 975 | RowBox[{"{", 976 | RowBox[{"0", ",", "1", ",", "0", ",", 977 | RowBox[{"-", "1"}], ",", "0", ",", "1", ",", "0"}], "}"}]], "Output", 978 | CellChangeTimes->{3.949928770509386*^9}, 979 | CellLabel->"Out[5]=",ExpressionUUID->"72083966-ad16-40ed-906a-0b32fda19dbc"] 980 | }, Open ]], 981 | 982 | Cell[TextData[{ 983 | "In (9) we substitute all the results, starting with the fact that the sine \ 984 | of ", 985 | Cell[BoxData[ 986 | FormBox["0", TraditionalForm]],ExpressionUUID-> 987 | "f5a7bc76-ceae-4fa9-befc-8e0c73b71932"], 988 | " is ", 989 | Cell[BoxData[ 990 | FormBox["0", TraditionalForm]],ExpressionUUID-> 991 | "0e697515-318c-4950-bce8-0009a128857d"], 992 | ". " 993 | }], "Text", 994 | CellChangeTimes->{{3.9469127456920843`*^9, 3.946912758168522*^9}, { 995 | 3.94691284850578*^9, 3.94691286841567*^9}, {3.949397721967601*^9, 996 | 3.949397732886392*^9}},ExpressionUUID->"b34734be-ca68-4e4f-96e0-\ 997 | a098c533b3e5"], 998 | 999 | Cell[BoxData[{ 1000 | RowBox[{ 1001 | RowBox[{"f", 1002 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1003 | RowBox[{"sin", 1004 | RowBox[{"(", "\[Theta]", ")"}]}]}], "\[IndentingNewLine]", 1005 | RowBox[{ 1006 | RowBox[{"sin", 1007 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1008 | RowBox[{ 1009 | RowBox[{ 1010 | FractionBox[ 1011 | RowBox[{"f", 1012 | RowBox[{"(", "0", ")"}]}], 1013 | RowBox[{"0", "!"}]], 1014 | SuperscriptBox["\[Theta]", "0"]}], "+", 1015 | RowBox[{ 1016 | FractionBox[ 1017 | RowBox[{ 1018 | RowBox[{"f", "'"}], 1019 | RowBox[{"(", "0", ")"}]}], 1020 | RowBox[{"1", "!"}]], 1021 | SuperscriptBox["\[Theta]", "1"]}], "+", 1022 | RowBox[{ 1023 | FractionBox[ 1024 | RowBox[{ 1025 | RowBox[{"f", "''"}], 1026 | RowBox[{"(", "0", ")"}]}], 1027 | RowBox[{"2", "!"}]], 1028 | SuperscriptBox["\[Theta]", "2"]}], "+", 1029 | RowBox[{ 1030 | FractionBox[ 1031 | RowBox[{ 1032 | SuperscriptBox["f", 1033 | RowBox[{"(", "3", ")"}]], 1034 | RowBox[{"(", "0", ")"}]}], 1035 | RowBox[{"3", "!"}]], 1036 | SuperscriptBox["\[Theta]", "3"]}], "+", "\[Ellipsis]"}]}], "\[IndentingNewLine]", 1037 | RowBox[{ 1038 | RowBox[{"sin", 1039 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1040 | RowBox[{ 1041 | RowBox[{ 1042 | FractionBox["0", 1043 | RowBox[{"0", "!"}]], 1044 | SuperscriptBox["\[Theta]", "0"]}], "+", 1045 | RowBox[{ 1046 | FractionBox["1", 1047 | RowBox[{"1", "!"}]], "\[Theta]"}], "+", 1048 | RowBox[{ 1049 | FractionBox["0", 1050 | RowBox[{"2", "!"}]], 1051 | SuperscriptBox["\[Theta]", "2"]}], "+", 1052 | RowBox[{ 1053 | FractionBox[ 1054 | RowBox[{"-", "1"}], 1055 | RowBox[{"3", "!"}]], 1056 | SuperscriptBox["\[Theta]", "3"]}], "+", 1057 | RowBox[{ 1058 | FractionBox["0", 1059 | RowBox[{"4", "!"}]], 1060 | SuperscriptBox["\[Theta]", "4"]}], "+", 1061 | RowBox[{ 1062 | FractionBox["1", 1063 | RowBox[{"5", "!"}]], 1064 | SuperscriptBox["\[Theta]", "5"]}], "+", 1065 | RowBox[{ 1066 | FractionBox["0", 1067 | RowBox[{"6", "!"}]], 1068 | SuperscriptBox["\[Theta]", "6"]}], "+", "\[Ellipsis]"}]}], "\[IndentingNewLine]", 1069 | RowBox[{ 1070 | RowBox[{"sin", 1071 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1072 | RowBox[{"\[Theta]", "-", 1073 | FractionBox[ 1074 | SuperscriptBox["\[Theta]", "3"], 1075 | RowBox[{"3", "!"}]], "+", 1076 | RowBox[{ 1077 | FractionBox[ 1078 | SuperscriptBox["\[Theta]", "5"], 1079 | RowBox[{"5", "!"}]], "\[Ellipsis]"}]}]}]}], "DisplayFormulaNumbered", 1080 | CellChangeTimes->{{3.946912398753281*^9, 3.946912401835047*^9}, { 1081 | 3.946912489011959*^9, 3.946912604629195*^9}, {3.9469128783657503`*^9, 1082 | 3.9469130013477573`*^9}, {3.9469130825259323`*^9, 3.946913127804245*^9}, { 1083 | 3.949396664283824*^9, 3.94939667110186*^9}, {3.9493977361196423`*^9, 1084 | 3.949397788710144*^9}, {3.949397841393659*^9, 1085 | 3.94939787918511*^9}},ExpressionUUID->"deeadaaf-6df4-40a0-beb4-\ 1086 | 1ba9435eee18"], 1087 | 1088 | Cell[CellGroupData[{ 1089 | 1090 | Cell[BoxData[ 1091 | RowBox[{ 1092 | RowBox[{"(*", 1093 | RowBox[{ 1094 | "Approximate", " ", "the", " ", "sine", " ", "function", " ", "around", " ", 1095 | Cell["\[Theta]=0",ExpressionUUID-> 1096 | "6ac54d6e-80b6-44dd-9695-d1ae30572ee2"], " ", "with", " ", "more", " ", 1097 | "and", " ", "more", " ", "terms"}], "*)"}], "\[IndentingNewLine]", 1098 | RowBox[{"Manipulate", "[", 1099 | RowBox[{ 1100 | RowBox[{"Normal", "[", 1101 | RowBox[{"Evaluate", "[", 1102 | RowBox[{"Series", "[", 1103 | RowBox[{ 1104 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 1105 | RowBox[{"{", 1106 | RowBox[{"\[Theta]", ",", "0", ",", "w"}], "}"}]}], "]"}], "]"}], 1107 | "]"}], ",", 1108 | RowBox[{"{", 1109 | RowBox[{"w", ",", "2", ",", "24", ",", "2"}], "}"}]}], "]"}]}]], "Input",\ 1110 | 1111 | CellChangeTimes->{{3.946912659776814*^9, 3.946912661036429*^9}, { 1112 | 3.946913559409396*^9, 3.946913563918643*^9}}, 1113 | CellLabel->"In[6]:=",ExpressionUUID->"aeb74690-ca2d-4648-a996-1eb1860fe6ac"], 1114 | 1115 | Cell[BoxData[ 1116 | TagBox[ 1117 | StyleBox[ 1118 | DynamicModuleBox[{$CellContext`w$$ = 24, Typeset`show$$ = True, 1119 | Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 1120 | Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 1121 | "\"untitled\"", Typeset`specs$$ = {{ 1122 | Hold[$CellContext`w$$], 2, 24, 2, ControlType -> Manipulator}}, 1123 | Typeset`size$$ = { 1124 | 1058.1857128906245`, {69.1265660546875, 29.283405195312504`}}, 1125 | Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, 1126 | Typeset`keyframeActionsQ$$ = False, Typeset`keyframeList$$ = {}}, 1127 | DynamicBox[Manipulate`ManipulateBoxes[ 1128 | 1, StandardForm, "Variables" :> {$CellContext`w$$ = 2}, 1129 | "ControllerVariables" :> {}, 1130 | "OtherVariables" :> { 1131 | Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 1132 | Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 1133 | Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, 1134 | Typeset`skipInitDone$$, Typeset`keyframeActionsQ$$, 1135 | Typeset`keyframeList$$}, "Body" :> Normal[ 1136 | Evaluate[ 1137 | Series[ 1138 | 1139 | Sin[$CellContext`\[Theta]], {$CellContext`\[Theta], 1140 | 0, $CellContext`w$$}]]], 1141 | "Specifications" :> {{$CellContext`w$$, 2, 24, 2}}, "Options" :> {}, 1142 | "DefaultOptions" :> {}], 1143 | ImageSizeCache->{1744.08, {114.4116735522461, 123.0095739428711}}, 1144 | SingleEvaluation->True], 1145 | Deinitialization:>None, 1146 | DynamicModuleValues:>{}, 1147 | SynchronousInitialization->True, 1148 | UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, 1149 | UnsavedVariables:>{Typeset`initDone$$}, 1150 | UntrackedVariables:>{Typeset`size$$}], "Manipulate", 1151 | Deployed->True, 1152 | StripOnInput->False], 1153 | Manipulate`InterpretManipulate[1]]], "Output", 1154 | CellChangeTimes->{3.949928810526751*^9}, 1155 | CellLabel->"Out[6]=",ExpressionUUID->"664a669a-27e8-489b-8679-0d7dbda1b40c"] 1156 | }, Open ]], 1157 | 1158 | Cell["\<\ 1159 | As before, the more terms we use, the better the approximation.\ 1160 | \>", "Text", 1161 | CellChangeTimes->{{3.94691341205157*^9, 1162 | 3.9469134274894247`*^9}},ExpressionUUID->"1247afad-fe42-4f0f-971f-\ 1163 | ccfa094a6c23"], 1164 | 1165 | Cell[CellGroupData[{ 1166 | 1167 | Cell[BoxData[ 1168 | RowBox[{ 1169 | RowBox[{"(*", 1170 | RowBox[{ 1171 | "Approximate", " ", "the", " ", "sine", " ", "function", " ", "around", " ", 1172 | Cell["\[Theta]=0",ExpressionUUID-> 1173 | "60c21489-c60c-4470-9f21-fa7e3c1dc227"], " ", "with", " ", "an", " ", 1174 | "increasing", " ", "number", " ", "of", " ", "terms"}], "*)"}], 1175 | "\[IndentingNewLine]", 1176 | RowBox[{"Manipulate", "[", 1177 | RowBox[{ 1178 | RowBox[{"Column", "[", 1179 | RowBox[{"{", 1180 | RowBox[{ 1181 | RowBox[{"Normal", "[", 1182 | RowBox[{"Evaluate", "[", 1183 | RowBox[{"Series", "[", 1184 | RowBox[{ 1185 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 1186 | RowBox[{"{", 1187 | RowBox[{"\[Theta]", ",", "0", ",", "w"}], "}"}]}], "]"}], "]"}], 1188 | "]"}], ",", 1189 | RowBox[{"Plot", "[", 1190 | RowBox[{ 1191 | RowBox[{"{", 1192 | RowBox[{ 1193 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 1194 | RowBox[{"Evaluate", "[", 1195 | RowBox[{"Normal", "[", 1196 | RowBox[{"Series", "[", 1197 | RowBox[{ 1198 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 1199 | RowBox[{"{", 1200 | RowBox[{"\[Theta]", ",", "0", ",", "w"}], "}"}]}], "]"}], 1201 | "]"}], "]"}]}], "}"}], ",", 1202 | RowBox[{"{", 1203 | RowBox[{"\[Theta]", ",", 1204 | RowBox[{ 1205 | RowBox[{"-", "5"}], " ", "\[Pi]"}], ",", 1206 | RowBox[{"5", " ", "\[Pi]"}]}], "}"}], ",", 1207 | RowBox[{"PlotRange", "->", 1208 | RowBox[{"{", 1209 | RowBox[{ 1210 | RowBox[{"{", 1211 | RowBox[{ 1212 | RowBox[{ 1213 | RowBox[{"-", "5"}], " ", "\[Pi]"}], ",", 1214 | RowBox[{"5", " ", "\[Pi]"}]}], "}"}], ",", 1215 | RowBox[{"{", 1216 | RowBox[{ 1217 | RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "}"}]}], ",", 1218 | RowBox[{"GridLines", "->", "Automatic"}], ",", 1219 | RowBox[{"ImageSize", "->", "Large"}]}], "]"}]}], "}"}], "]"}], ",", 1220 | RowBox[{"{", 1221 | RowBox[{"w", ",", "2", ",", "24", ",", "2"}], "}"}]}], "]"}]}]], "Input",\ 1222 | 1223 | CellChangeTimes->{{3.946908198186825*^9, 3.946908403468902*^9}, { 1224 | 3.946908434883709*^9, 3.946908484116255*^9}, {3.946908630784532*^9, 1225 | 3.946908640668055*^9}, {3.946913573352202*^9, 3.9469135776883287`*^9}, { 1226 | 3.9493970806132517`*^9, 3.949397088358296*^9}, {3.949397120934887*^9, 1227 | 3.949397127762643*^9}}, 1228 | CellLabel->"In[7]:=",ExpressionUUID->"7fff56ac-3664-4d75-ad94-681d3feab7c5"], 1229 | 1230 | Cell[BoxData[ 1231 | TagBox[ 1232 | StyleBox[ 1233 | DynamicModuleBox[{$CellContext`w$$ = 24, Typeset`show$$ = True, 1234 | Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", 1235 | Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = 1236 | "\"untitled\"", Typeset`specs$$ = {{ 1237 | Hold[$CellContext`w$$], 2, 24, 2, ControlType -> Manipulator}}, 1238 | Typeset`size$$ = { 1239 | 1414.348403613281, {290.94023570870473`, 299.53813609932973`}}, 1240 | Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, 1241 | Typeset`keyframeActionsQ$$ = False, Typeset`keyframeList$$ = {}}, 1242 | DynamicBox[Manipulate`ManipulateBoxes[ 1243 | 1, StandardForm, "Variables" :> {$CellContext`w$$ = 2}, 1244 | "ControllerVariables" :> {}, 1245 | "OtherVariables" :> { 1246 | Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, 1247 | Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, 1248 | Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, 1249 | Typeset`skipInitDone$$, Typeset`keyframeActionsQ$$, 1250 | Typeset`keyframeList$$}, "Body" :> Column[{ 1251 | Normal[ 1252 | Evaluate[ 1253 | Series[ 1254 | 1255 | Sin[$CellContext`\[Theta]], {$CellContext`\[Theta], 1256 | 0, $CellContext`w$$}]]], 1257 | Plot[{ 1258 | Sin[$CellContext`\[Theta]], 1259 | Evaluate[ 1260 | Normal[ 1261 | Series[ 1262 | 1263 | Sin[$CellContext`\[Theta]], {$CellContext`\[Theta], 1264 | 0, $CellContext`w$$}]]]}, {$CellContext`\[Theta], (-5) Pi, 5 1265 | Pi}, PlotRange -> {{(-5) Pi, 5 Pi}, {-2, 2}}, GridLines -> 1266 | Automatic, ImageSize -> Large]}], 1267 | "Specifications" :> {{$CellContext`w$$, 2, 24, 2}}, "Options" :> {}, 1268 | "DefaultOptions" :> {}], 1269 | ImageSizeCache->{ 1270 | 1483.9984036132812`, {360.4458738312633, 369.0437742218883}}, 1271 | SingleEvaluation->True], 1272 | Deinitialization:>None, 1273 | DynamicModuleValues:>{}, 1274 | SynchronousInitialization->True, 1275 | UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$}, 1276 | UnsavedVariables:>{Typeset`initDone$$}, 1277 | UntrackedVariables:>{Typeset`size$$}], "Manipulate", 1278 | Deployed->True, 1279 | StripOnInput->False], 1280 | Manipulate`InterpretManipulate[1]]], "Output", 1281 | CellChangeTimes->{3.9499288263359537`*^9}, 1282 | CellLabel->"Out[7]=",ExpressionUUID->"2fe809ca-0425-466c-b557-89220fd2ac41"] 1283 | }, Open ]], 1284 | 1285 | Cell[TextData[{ 1286 | "In the code cell below, we add the terms of the Maclaurin series of the \ 1287 | cosine and the sine function. The ", 1288 | StyleBox["Normal", "CodeText"], 1289 | " function returns an expression from the list object that the ", 1290 | StyleBox["Series", "CodeText"], 1291 | " function returns." 1292 | }], "Text", 1293 | CellChangeTimes->{{3.9469136076978893`*^9, 3.946913635909669*^9}, { 1294 | 3.946913798506865*^9, 1295 | 3.946913824042897*^9}},ExpressionUUID->"d0348224-a8af-4533-9aef-\ 1296 | 7e21568e91f0"] 1297 | }, Open ]], 1298 | 1299 | Cell[CellGroupData[{ 1300 | 1301 | Cell["Deriving the Euler form of a complex number", "Section", 1302 | CellChangeTimes->{{3.949397904316071*^9, 3.949397912938863*^9}, { 1303 | 3.9494294878925323`*^9, 1304 | 3.949429490563599*^9}},ExpressionUUID->"f043eb99-7f58-4e23-b4e4-\ 1305 | eed166b4fbaf"], 1306 | 1307 | Cell[CellGroupData[{ 1308 | 1309 | Cell[BoxData[ 1310 | RowBox[{ 1311 | RowBox[{"(*", 1312 | RowBox[{ 1313 | "Add", " ", "the", " ", "terms", " ", "of", " ", "the", " ", "Maclaurin", " ", 1314 | "series", " ", "of", " ", "the", " ", "cosine", " ", "and", " ", "the", " ", 1315 | "sine", " ", "functions", " ", "and", " ", "use", " ", "the", " ", 1316 | "Normal", " ", "function", " ", "to", " ", "return", " ", "an", " ", 1317 | "expression"}], "*)"}], "\[IndentingNewLine]", 1318 | RowBox[{"Normal", "[", 1319 | RowBox[{ 1320 | RowBox[{"Series", "[", 1321 | RowBox[{ 1322 | RowBox[{"Cos", "[", "\[Theta]", "]"}], ",", 1323 | RowBox[{"{", 1324 | RowBox[{"\[Theta]", ",", "0", ",", "10"}], "}"}]}], "]"}], "+", 1325 | RowBox[{"Series", "[", 1326 | RowBox[{ 1327 | RowBox[{"Sin", "[", "\[Theta]", "]"}], ",", 1328 | RowBox[{"{", 1329 | RowBox[{"\[Theta]", ",", "0", ",", "10"}], "}"}]}], "]"}]}], 1330 | "]"}]}]], "Input", 1331 | CellChangeTimes->{{3.946908763639657*^9, 3.946908787174643*^9}, { 1332 | 3.946913641017398*^9, 3.946913700788807*^9}, 3.947518052178591*^9}, 1333 | CellLabel->"In[8]:=",ExpressionUUID->"4991ece7-daac-43b8-92e6-598d1b51b9fa"], 1334 | 1335 | Cell[BoxData[ 1336 | RowBox[{"1", "+", "\[Theta]", "-", 1337 | FractionBox[ 1338 | SuperscriptBox["\[Theta]", "2"], "2"], "-", 1339 | FractionBox[ 1340 | SuperscriptBox["\[Theta]", "3"], "6"], "+", 1341 | FractionBox[ 1342 | SuperscriptBox["\[Theta]", "4"], "24"], "+", 1343 | FractionBox[ 1344 | SuperscriptBox["\[Theta]", "5"], "120"], "-", 1345 | FractionBox[ 1346 | SuperscriptBox["\[Theta]", "6"], "720"], "-", 1347 | FractionBox[ 1348 | SuperscriptBox["\[Theta]", "7"], "5040"], "+", 1349 | FractionBox[ 1350 | SuperscriptBox["\[Theta]", "8"], "40320"], "+", 1351 | FractionBox[ 1352 | SuperscriptBox["\[Theta]", "9"], "362880"], "-", 1353 | FractionBox[ 1354 | SuperscriptBox["\[Theta]", "10"], "3628800"]}]], "Output", 1355 | CellChangeTimes->{3.9499288701096497`*^9}, 1356 | CellLabel->"Out[8]=",ExpressionUUID->"10cc96f5-ea60-49e9-9c16-17da30e2a3fd"] 1357 | }, Open ]], 1358 | 1359 | Cell[TextData[{ 1360 | "This looks very similar to the Maclaurin series of the exponent ", 1361 | Cell[BoxData[ 1362 | FormBox[ 1363 | SuperscriptBox["e", "\[Theta]"], TraditionalForm]],ExpressionUUID-> 1364 | "5977062b-b419-4dba-88c1-6202aee2fd2c"], 1365 | ", which we express below." 1366 | }], "Text", 1367 | CellChangeTimes->{{3.9469137235318737`*^9, 1368 | 3.946913754912766*^9}},ExpressionUUID->"5526ed6a-bfce-470f-b85f-\ 1369 | 6eed436d44a9"], 1370 | 1371 | Cell[CellGroupData[{ 1372 | 1373 | Cell[BoxData[ 1374 | RowBox[{ 1375 | RowBox[{"(*", 1376 | RowBox[{ 1377 | "Show", " ", "the", " ", "Maclaurin", " ", "series", " ", "of", " ", "the", 1378 | " ", "exponent", " ", "function", " ", "and", " ", "use", " ", "the", " ", 1379 | "Normal", " ", "function", " ", "to", " ", "return", " ", "an", " ", 1380 | "expression"}], "*)"}], "\[IndentingNewLine]", 1381 | RowBox[{"Normal", "[", 1382 | RowBox[{"Series", "[", 1383 | RowBox[{ 1384 | SuperscriptBox["\[ExponentialE]", "\[Theta]"], ",", 1385 | RowBox[{"{", 1386 | RowBox[{"\[Theta]", ",", "0", ",", "10"}], "}"}]}], "]"}], 1387 | "]"}]}]], "Input", 1388 | CellChangeTimes->{{3.946908809606125*^9, 3.9469088097741413`*^9}, { 1389 | 3.9469137583093777`*^9, 3.9469137958497143`*^9}}, 1390 | CellLabel->"In[9]:=",ExpressionUUID->"ac8e91a1-6437-4cfa-a94b-d9b3447f856d"], 1391 | 1392 | Cell[BoxData[ 1393 | RowBox[{"1", "+", "\[Theta]", "+", 1394 | FractionBox[ 1395 | SuperscriptBox["\[Theta]", "2"], "2"], "+", 1396 | FractionBox[ 1397 | SuperscriptBox["\[Theta]", "3"], "6"], "+", 1398 | FractionBox[ 1399 | SuperscriptBox["\[Theta]", "4"], "24"], "+", 1400 | FractionBox[ 1401 | SuperscriptBox["\[Theta]", "5"], "120"], "+", 1402 | FractionBox[ 1403 | SuperscriptBox["\[Theta]", "6"], "720"], "+", 1404 | FractionBox[ 1405 | SuperscriptBox["\[Theta]", "7"], "5040"], "+", 1406 | FractionBox[ 1407 | SuperscriptBox["\[Theta]", "8"], "40320"], "+", 1408 | FractionBox[ 1409 | SuperscriptBox["\[Theta]", "9"], "362880"], "+", 1410 | FractionBox[ 1411 | SuperscriptBox["\[Theta]", "10"], "3628800"]}]], "Output", 1412 | CellChangeTimes->{3.949928883477254*^9}, 1413 | CellLabel->"Out[9]=",ExpressionUUID->"247ba9e8-7542-4712-a992-e364678c3c1d"] 1414 | }, Open ]], 1415 | 1416 | Cell[TextData[{ 1417 | "The series expansion of ", 1418 | Cell[BoxData[ 1419 | FormBox[ 1420 | SuperscriptBox["e", "\[Theta]"], TraditionalForm]],ExpressionUUID-> 1421 | "fb59c468-18ca-4001-81fe-69cbe9cbe56a"], 1422 | ", shown in (10), is very simple since all derivatives are simply ", 1423 | Cell[BoxData[ 1424 | FormBox[ 1425 | SuperscriptBox["e", "\[Theta]"], TraditionalForm]],ExpressionUUID-> 1426 | "584432f3-97cf-4c65-95ff-90954ba91a49"], 1427 | "." 1428 | }], "Text", 1429 | CellChangeTimes->{{3.94691398077168*^9, 3.946914024824004*^9}, { 1430 | 3.949397248634101*^9, 3.94939725149389*^9}, {3.949397935053338*^9, 1431 | 3.949397952109479*^9}},ExpressionUUID->"8c9a5b71-4887-45d0-8740-\ 1432 | 75569fb37e83"], 1433 | 1434 | Cell[BoxData[{ 1435 | RowBox[{ 1436 | RowBox[{"f", 1437 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1438 | SuperscriptBox["e", "\[Theta]"]}], "\[IndentingNewLine]", 1439 | RowBox[{ 1440 | RowBox[{ 1441 | RowBox[{"f", "'"}], 1442 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1443 | SuperscriptBox["e", 1444 | "\[Theta]"]}], "\[IndentingNewLine]", "\[VerticalEllipsis]"}], \ 1445 | "DisplayFormulaNumbered", 1446 | CellChangeTimes->{{3.949397188703496*^9, 1447 | 3.949397220607333*^9}},ExpressionUUID->"5aba99b3-7ba1-493b-835a-\ 1448 | 0fc6346c8723"], 1449 | 1450 | Cell[TextData[{ 1451 | "Instead of ", 1452 | Cell[BoxData[ 1453 | FormBox[ 1454 | SuperscriptBox["e", "\[Theta]"], TraditionalForm]],ExpressionUUID-> 1455 | "c2f1bbf7-58cf-4e28-8f5d-a3cdf33f6819"], 1456 | ", we can take a look at the series expansion of ", 1457 | Cell[BoxData[ 1458 | FormBox[ 1459 | SuperscriptBox["e", "\[Alpha]\[Theta]"], TraditionalForm]],ExpressionUUID-> 1460 | "6ed497de-a110-4e97-8ae2-ac3e0fd85ae7"], 1461 | ", where we multiply ", 1462 | Cell[BoxData[ 1463 | FormBox["\[Theta]", TraditionalForm]],ExpressionUUID-> 1464 | "477edf12-7398-4a86-9052-243204063624"], 1465 | " by a constant ", 1466 | Cell[BoxData[ 1467 | FormBox["\[Alpha]", TraditionalForm]],ExpressionUUID-> 1468 | "aa01da39-8329-4ca1-9a45-c0d3ae6658a1"], 1469 | ", shown in (11)." 1470 | }], "Text", 1471 | CellChangeTimes->{{3.946913856861553*^9, 3.9469139048605967`*^9}, { 1472 | 3.946914034870529*^9, 3.946914036862276*^9}, {3.949397257241673*^9, 1473 | 3.949397259528541*^9}, {3.94939795771809*^9, 1474 | 3.9493979699476013`*^9}},ExpressionUUID->"ea937e02-3b56-4786-bbd0-\ 1475 | 4b6d87b656a4"], 1476 | 1477 | Cell[BoxData[{ 1478 | RowBox[{ 1479 | RowBox[{"f", 1480 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1481 | SuperscriptBox["e", "\[Alpha]\[Theta]"]}], "\[IndentingNewLine]", 1482 | RowBox[{ 1483 | RowBox[{ 1484 | RowBox[{"f", "'"}], 1485 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1486 | SuperscriptBox["\[Alpha]e", "\[Alpha]\[Theta]"]}], "\[IndentingNewLine]", 1487 | RowBox[{ 1488 | RowBox[{ 1489 | RowBox[{"f", "''"}], 1490 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1491 | RowBox[{ 1492 | SuperscriptBox["\[Alpha]", "2"], 1493 | SuperscriptBox["e", "\[Alpha]\[Theta]"]}]}], "\[IndentingNewLine]", 1494 | RowBox[{ 1495 | RowBox[{ 1496 | SuperscriptBox["f", 1497 | RowBox[{"(", "3", ")"}]], 1498 | RowBox[{"(", "\[Theta]", ")"}]}], "=", 1499 | RowBox[{ 1500 | SuperscriptBox["\[Alpha]", "3"], 1501 | SuperscriptBox["e", 1502 | "\[Alpha]\[Theta]"]}]}], "\[IndentingNewLine]", "\[VerticalEllipsis]"}], \ 1503 | "DisplayFormulaNumbered", 1504 | CellChangeTimes->{{3.949397292622861*^9, 1505 | 3.9493973450969687`*^9}},ExpressionUUID->"f18db181-8b31-4cca-834b-\ 1506 | 98caa4a2faf7"], 1507 | 1508 | Cell[TextData[{ 1509 | "We verify the series expansion of ", 1510 | Cell[BoxData[ 1511 | FormBox[ 1512 | TemplateBox[<|"boxes" -> FormBox[ 1513 | SuperscriptBox[ 1514 | StyleBox["e", "TI"], "\[Alpha]\[Theta]"], TraditionalForm], 1515 | "errors" -> {}, "input" -> "e^{\\alpha \\theta}", "state" -> "Boxes"|>, 1516 | "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> 1517 | "fdabb61f-c2d7-4494-861c-0e333816ad5d"], 1518 | " using the ", 1519 | StyleBox["Series", "CodeText"], 1520 | " function." 1521 | }], "Text", 1522 | CellChangeTimes->{{3.949397265221838*^9, 3.9493972764241*^9}, { 1523 | 3.949397370589933*^9, 3.9493974030500593`*^9}, {3.9493979853622513`*^9, 1524 | 3.949397986845951*^9}},ExpressionUUID->"4da675c3-2c29-4d91-9fc9-\ 1525 | 1325c4a45aa0"], 1526 | 1527 | Cell[CellGroupData[{ 1528 | 1529 | Cell[BoxData[ 1530 | RowBox[{ 1531 | RowBox[{"(*", 1532 | RowBox[{ 1533 | "Show", " ", "the", " ", "series", " ", "expansion", " ", "of", " ", Cell[ 1534 | TextData[Cell[BoxData[ 1535 | FormBox[ 1536 | SuperscriptBox["e", "\[Alpha]\[Theta]"], TraditionalForm]], 1537 | ExpressionUUID->"0f8547e1-80d4-4be5-806b-cfec1bc98412"]],ExpressionUUID-> 1538 | "29ae22de-7324-4bc4-a59f-5720f5fcb717"], " ", "and", " ", "use", " ", 1539 | "the", " ", "Normal", " ", "function", " ", "to", " ", "return", " ", 1540 | "an", " ", "expression"}], "*)"}], "\[IndentingNewLine]", 1541 | RowBox[{"Normal", "[", 1542 | RowBox[{"Series", "[", 1543 | RowBox[{ 1544 | SuperscriptBox["\[ExponentialE]", 1545 | RowBox[{"\[Alpha]", " ", "\[Theta]"}]], ",", 1546 | RowBox[{"{", 1547 | RowBox[{"\[Theta]", ",", "0", ",", "10"}], "}"}]}], "]"}], 1548 | "]"}]}]], "Input", 1549 | CellChangeTimes->{{3.9469085432701178`*^9, 3.946908567603189*^9}, { 1550 | 3.946913916992551*^9, 3.946913934383465*^9}, {3.949397992537435*^9, 1551 | 3.9493979972808332`*^9}}, 1552 | CellLabel->"In[10]:=",ExpressionUUID->"68ab044b-302e-4312-bde5-53ab17f47edf"], 1553 | 1554 | Cell[BoxData[ 1555 | RowBox[{"1", "+", 1556 | RowBox[{"\[Alpha]", " ", "\[Theta]"}], "+", 1557 | FractionBox[ 1558 | RowBox[{ 1559 | SuperscriptBox["\[Alpha]", "2"], " ", 1560 | SuperscriptBox["\[Theta]", "2"]}], "2"], "+", 1561 | FractionBox[ 1562 | RowBox[{ 1563 | SuperscriptBox["\[Alpha]", "3"], " ", 1564 | SuperscriptBox["\[Theta]", "3"]}], "6"], "+", 1565 | FractionBox[ 1566 | RowBox[{ 1567 | SuperscriptBox["\[Alpha]", "4"], " ", 1568 | SuperscriptBox["\[Theta]", "4"]}], "24"], "+", 1569 | FractionBox[ 1570 | RowBox[{ 1571 | SuperscriptBox["\[Alpha]", "5"], " ", 1572 | SuperscriptBox["\[Theta]", "5"]}], "120"], "+", 1573 | FractionBox[ 1574 | RowBox[{ 1575 | SuperscriptBox["\[Alpha]", "6"], " ", 1576 | SuperscriptBox["\[Theta]", "6"]}], "720"], "+", 1577 | FractionBox[ 1578 | RowBox[{ 1579 | SuperscriptBox["\[Alpha]", "7"], " ", 1580 | SuperscriptBox["\[Theta]", "7"]}], "5040"], "+", 1581 | FractionBox[ 1582 | RowBox[{ 1583 | SuperscriptBox["\[Alpha]", "8"], " ", 1584 | SuperscriptBox["\[Theta]", "8"]}], "40320"], "+", 1585 | FractionBox[ 1586 | RowBox[{ 1587 | SuperscriptBox["\[Alpha]", "9"], " ", 1588 | SuperscriptBox["\[Theta]", "9"]}], "362880"], "+", 1589 | FractionBox[ 1590 | RowBox[{ 1591 | SuperscriptBox["\[Alpha]", "10"], " ", 1592 | SuperscriptBox["\[Theta]", "10"]}], "3628800"]}]], "Output", 1593 | CellChangeTimes->{3.949928960114706*^9}, 1594 | CellLabel->"Out[10]=",ExpressionUUID->"eae00a9c-888d-49d4-bbb4-68bb561ca41e"] 1595 | }, Open ]], 1596 | 1597 | Cell[TextData[{ 1598 | "Since ", 1599 | Cell[BoxData[ 1600 | FormBox["\[Alpha]", TraditionalForm]],ExpressionUUID-> 1601 | "d49c7ba8-914d-4444-83ff-f0cf0e0e2f6d"], 1602 | " is a constant, we can replace it with a number such as the imaginary unit ", 1603 | Cell[BoxData[ 1604 | FormBox["i", TraditionalForm]],ExpressionUUID-> 1605 | "05fa8b2e-2393-46d5-aca6-bfcebcd2baef"], 1606 | "." 1607 | }], "Text", 1608 | CellChangeTimes->{{3.946914274585888*^9, 1609 | 3.946914293982127*^9}},ExpressionUUID->"54ca97b7-e0f3-4ed1-8e65-\ 1610 | 193603ab60f7"], 1611 | 1612 | Cell[CellGroupData[{ 1613 | 1614 | Cell[BoxData[ 1615 | RowBox[{ 1616 | RowBox[{"(*", 1617 | RowBox[{ 1618 | "Show", " ", "the", " ", "series", " ", "exansion", " ", "of", " ", Cell[ 1619 | TextData[Cell[BoxData[ 1620 | FormBox[ 1621 | SuperscriptBox["e", "i\[Theta]"], TraditionalForm]],ExpressionUUID-> 1622 | "132dcd2f-61b8-499d-9a3c-9b2630d35bd7"]],ExpressionUUID-> 1623 | "ae0fddc0-d74a-4aa2-99a7-786f41dec360"]}], "*)"}], "\[IndentingNewLine]", 1624 | RowBox[{"Normal", "[", 1625 | RowBox[{"Series", "[", 1626 | RowBox[{ 1627 | SuperscriptBox["\[ExponentialE]", 1628 | RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]], ",", 1629 | RowBox[{"{", 1630 | RowBox[{"\[Theta]", ",", "0", ",", "10"}], "}"}]}], "]"}], 1631 | "]"}]}]], "Input", 1632 | CellChangeTimes->{{3.946908594362741*^9, 3.946908612132779*^9}, { 1633 | 3.946914297660962*^9, 3.946914345799811*^9}}, 1634 | CellLabel->"In[11]:=",ExpressionUUID->"db2e1247-6f85-49a8-b09c-dbb285317218"], 1635 | 1636 | Cell[BoxData[ 1637 | RowBox[{"1", "+", 1638 | RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}], "-", 1639 | FractionBox[ 1640 | SuperscriptBox["\[Theta]", "2"], "2"], "-", 1641 | FractionBox[ 1642 | RowBox[{"\[ImaginaryI]", " ", 1643 | SuperscriptBox["\[Theta]", "3"]}], "6"], "+", 1644 | FractionBox[ 1645 | SuperscriptBox["\[Theta]", "4"], "24"], "+", 1646 | FractionBox[ 1647 | RowBox[{"\[ImaginaryI]", " ", 1648 | SuperscriptBox["\[Theta]", "5"]}], "120"], "-", 1649 | FractionBox[ 1650 | SuperscriptBox["\[Theta]", "6"], "720"], "-", 1651 | FractionBox[ 1652 | RowBox[{"\[ImaginaryI]", " ", 1653 | SuperscriptBox["\[Theta]", "7"]}], "5040"], "+", 1654 | FractionBox[ 1655 | SuperscriptBox["\[Theta]", "8"], "40320"], "+", 1656 | FractionBox[ 1657 | RowBox[{"\[ImaginaryI]", " ", 1658 | SuperscriptBox["\[Theta]", "9"]}], "362880"], "-", 1659 | FractionBox[ 1660 | SuperscriptBox["\[Theta]", "10"], "3628800"]}]], "Output", 1661 | CellChangeTimes->{3.9499289823072844`*^9}, 1662 | CellLabel->"Out[11]=",ExpressionUUID->"1823ebf5-aa1d-4f00-8e3e-268b4a1d10ec"] 1663 | }, Open ]], 1664 | 1665 | Cell["\<\ 1666 | In (12), we group the terms and notice that the expansion contains the \ 1667 | Maclaurin series of the cosine and the sine functions.\ 1668 | \>", "Text", 1669 | CellChangeTimes->{{3.94691437069403*^9, 3.9469144265853577`*^9}, { 1670 | 3.9493980245973263`*^9, 1671 | 3.9493980248763657`*^9}},ExpressionUUID->"6935125c-2c7c-45b3-b813-\ 1672 | 6774c2ade9ef"], 1673 | 1674 | Cell[BoxData[{ 1675 | RowBox[{ 1676 | SuperscriptBox["e", "i\[Theta]"], "\[TildeTilde]", 1677 | RowBox[{ 1678 | RowBox[{"(", 1679 | RowBox[{"1", "-", 1680 | FractionBox[ 1681 | SuperscriptBox["\[Theta]", "2"], 1682 | RowBox[{"2", "!"}]], "+", 1683 | FractionBox[ 1684 | SuperscriptBox["\[Theta]", "4"], 1685 | RowBox[{"4", "!"}]], "-", 1686 | FractionBox[ 1687 | SuperscriptBox["\[Theta]", "6"], 1688 | RowBox[{"6", "!"}]], "+", 1689 | FractionBox[ 1690 | SuperscriptBox["\[Theta]", "8"], 1691 | RowBox[{"8", "!"}]], "-", 1692 | FractionBox[ 1693 | SuperscriptBox["\[Theta]", "10"], 1694 | RowBox[{"10", "!"}]]}], ")"}], "+", 1695 | RowBox[{"i", 1696 | RowBox[{"(", 1697 | RowBox[{"\[Theta]", "-", 1698 | FractionBox[ 1699 | SuperscriptBox["\[Theta]", "3"], 1700 | RowBox[{"3", "!"}]], "+", 1701 | FractionBox[ 1702 | SuperscriptBox["\[Theta]", "5"], 1703 | RowBox[{"5", "!"}]], "-", 1704 | FractionBox[ 1705 | SuperscriptBox["\[Theta]", "7"], 1706 | RowBox[{"7", "!"}]], "+", 1707 | FractionBox[ 1708 | SuperscriptBox["\[Theta]", "9"], 1709 | RowBox[{"9", "!"}]]}], ")"}]}]}]}], "\[IndentingNewLine]", 1710 | RowBox[{ 1711 | SuperscriptBox["e", "i\[Theta]"], "=", 1712 | RowBox[{ 1713 | RowBox[{"(", 1714 | RowBox[{"1", "-", 1715 | FractionBox[ 1716 | SuperscriptBox["\[Theta]", "2"], 1717 | RowBox[{"2", "!"}]], "+", 1718 | FractionBox[ 1719 | SuperscriptBox["\[Theta]", "4"], 1720 | RowBox[{"4", "!"}]], "-", 1721 | FractionBox[ 1722 | SuperscriptBox["\[Theta]", "6"], 1723 | RowBox[{"6", "!"}]], "+", 1724 | FractionBox[ 1725 | SuperscriptBox["\[Theta]", "8"], 1726 | RowBox[{"8", "!"}]], "-", 1727 | FractionBox[ 1728 | SuperscriptBox["\[Theta]", "10"], 1729 | RowBox[{"10", "!"}]], "+", "\[Ellipsis]"}], ")"}], "+", 1730 | RowBox[{"i", 1731 | RowBox[{"(", 1732 | RowBox[{"\[Theta]", "-", 1733 | FractionBox[ 1734 | SuperscriptBox["\[Theta]", "3"], 1735 | RowBox[{"3", "!"}]], "+", 1736 | FractionBox[ 1737 | SuperscriptBox["\[Theta]", "5"], 1738 | RowBox[{"5", "!"}]], "-", 1739 | FractionBox[ 1740 | SuperscriptBox["\[Theta]", "7"], 1741 | RowBox[{"7", "!"}]], "+", "\[Ellipsis]"}], ")"}]}]}]}], "\[IndentingNewLine]", 1742 | RowBox[{ 1743 | SuperscriptBox["e", "i\[Theta]"], "=", 1744 | RowBox[{ 1745 | RowBox[{"cos", 1746 | RowBox[{"(", "\[Theta]", ")"}]}], "+", 1747 | RowBox[{"i", " ", "sin", 1748 | RowBox[{"(", "\[Theta]", ")"}]}]}]}], "\[IndentingNewLine]", 1749 | RowBox[{ 1750 | RowBox[{"r", " ", 1751 | SuperscriptBox["e", "i\[Theta]"]}], "=", 1752 | RowBox[{"r", "[", 1753 | RowBox[{ 1754 | RowBox[{"cos", 1755 | RowBox[{"(", "\[Theta]", ")"}]}], "+", 1756 | RowBox[{"i", " ", "sin", 1757 | RowBox[{"(", "\[Theta]", ")"}]}]}], "]"}]}], "\[IndentingNewLine]", 1758 | RowBox[{ 1759 | RowBox[{"r", " ", 1760 | SuperscriptBox["e", 1761 | RowBox[{"i", 1762 | RowBox[{"(", 1763 | RowBox[{"\[Theta]", "+", 1764 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}]]}], "=", 1765 | RowBox[{"r", "[", 1766 | RowBox[{ 1767 | RowBox[{"cos", 1768 | RowBox[{"(", 1769 | RowBox[{"\[Theta]", "+", 1770 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}], "+", 1771 | RowBox[{"i", " ", "sin", 1772 | RowBox[{"(", 1773 | RowBox[{"\[Theta]", "+", 1774 | RowBox[{"2", "\[Pi]", " ", "k"}]}], ")"}]}]}], 1775 | "]"}]}]}], "DisplayFormulaNumbered", 1776 | CellChangeTimes->{ 1777 | 3.946914437119454*^9, {3.949400400966136*^9, 3.949400407575453*^9}, { 1778 | 3.9494268068124933`*^9, 1779 | 3.9494268279389267`*^9}},ExpressionUUID->"5e3bc8b6-8d7d-48f8-b381-\ 1780 | 543f1e1e3a52"] 1781 | }, Open ]] 1782 | }, Open ]] 1783 | }, 1784 | WindowSize->{1920, 1172}, 1785 | WindowMargins->{{1792, Automatic}, {Automatic, 0}}, 1786 | Magnification:>1.5 Inherited, 1787 | FrontEndVersion->"14.2 for Mac OS X x86 (64-bit) (December 26, 2024)", 1788 | StyleDefinitions->"Default.nb", 1789 | ExpressionUUID->"ae14531a-8820-44ad-a4ab-d947a6eec703" 1790 | ] 1791 | (* End of Notebook Content *) 1792 | 1793 | (* Internal cache information *) 1794 | (*CellTagsOutline 1795 | CellTagsIndex->{} 1796 | *) 1797 | (*CellTagsIndex 1798 | CellTagsIndex->{} 1799 | *) 1800 | (*NotebookFileOutline 1801 | Notebook[{ 1802 | Cell[CellGroupData[{ 1803 | Cell[576, 22, 247, 4, 144, "Title",ExpressionUUID->"6301cd3f-0fd2-429a-9871-db5dacb69af2"], 1804 | Cell[CellGroupData[{ 1805 | Cell[848, 30, 223, 4, 99, "Section",ExpressionUUID->"386b3412-cb3f-4c65-80ca-4255d3a3917e"], 1806 | Cell[1074, 36, 1121, 33, 71, "DisplayFormulaNumbered",ExpressionUUID->"f8d80c33-6b09-4cd3-9fef-6718ca7fbd25"] 1807 | }, Open ]], 1808 | Cell[CellGroupData[{ 1809 | Cell[2232, 74, 181, 3, 99, "Section",ExpressionUUID->"08fe1613-aa92-4f93-b47c-fbef6d46e9a6"], 1810 | Cell[2416, 79, 953, 22, 50, "Text",ExpressionUUID->"9bccb5e1-436c-4383-b408-e07656fdf6ef"], 1811 | Cell[3372, 103, 1063, 28, 77, "DisplayFormulaNumbered",ExpressionUUID->"2a2029d9-3573-415d-bde0-45d3680b5777"] 1812 | }, Open ]], 1813 | Cell[CellGroupData[{ 1814 | Cell[4472, 136, 173, 3, 99, "Section",ExpressionUUID->"7cd5c749-8cfa-4366-a774-5b010afbe073"], 1815 | Cell[4648, 141, 997, 25, 50, "Text",ExpressionUUID->"b8df313e-6ca1-4a83-bebf-39b1523ce039"], 1816 | Cell[5648, 168, 1649, 56, 137, "DisplayFormulaNumbered",ExpressionUUID->"48b3c032-438b-46a9-bf84-ccdd7e316a6c"], 1817 | Cell[7300, 226, 694, 18, 50, "Text",ExpressionUUID->"fa6cafa4-d884-4878-bdff-1b175d8afaf9"], 1818 | Cell[7997, 246, 1661, 56, 269, "DisplayFormulaNumbered",ExpressionUUID->"222b91e2-00ff-4d0f-9b91-752984836119"], 1819 | Cell[9661, 304, 600, 14, 50, "Text",ExpressionUUID->"c5fe48d0-04f3-4c6e-9900-79eb15c4ceb5"], 1820 | Cell[CellGroupData[{ 1821 | Cell[10286, 322, 1080, 23, 74, "Input",ExpressionUUID->"c4f6590d-e7d5-4414-938d-30b83f0c4233"], 1822 | Cell[11369, 347, 543, 14, 50, "Output",ExpressionUUID->"271552bc-b193-4c0b-a0f6-d55a22469680"] 1823 | }, Open ]], 1824 | Cell[11927, 364, 508, 13, 50, "Text",ExpressionUUID->"4bb1173f-116c-4e12-b8a7-15950abc9b0f"], 1825 | Cell[12438, 379, 1779, 64, 269, "DisplayFormulaNumbered",ExpressionUUID->"c1f4d054-59e7-4be1-a69a-8ad96b666a54"], 1826 | Cell[14220, 445, 586, 14, 50, "Text",ExpressionUUID->"e757bd7a-9b65-4c7a-a3cc-d9e0b0dd6a42"], 1827 | Cell[CellGroupData[{ 1828 | Cell[14831, 463, 1053, 24, 74, "Input",ExpressionUUID->"7a1301b6-6e8d-4e04-96e0-948cdedef374"], 1829 | Cell[15887, 489, 281, 6, 50, "Output",ExpressionUUID->"6c28df34-0261-44fe-a6c6-fa887da9fe0c"] 1830 | }, Open ]], 1831 | Cell[16183, 498, 520, 15, 50, "Text",ExpressionUUID->"a446de9c-651a-49da-b3cb-052d4acd188e"], 1832 | Cell[16706, 515, 2604, 89, 204, "DisplayFormulaNumbered",ExpressionUUID->"871678df-4a10-4c39-ad62-8d744e85e6f6"], 1833 | Cell[19313, 606, 446, 9, 50, "Text",ExpressionUUID->"79890846-eb15-442b-8421-b6d79c93afb4"], 1834 | Cell[CellGroupData[{ 1835 | Cell[19784, 619, 1084, 25, 74, "Input",ExpressionUUID->"c05b281d-b8bc-42a5-b677-5c18c40bfa78"], 1836 | Cell[20871, 646, 1931, 41, 208, "Output",ExpressionUUID->"cc2d7473-fd1f-46d5-bbb7-85ef1af0af8a"] 1837 | }, Open ]], 1838 | Cell[22817, 690, 376, 8, 50, "Text",ExpressionUUID->"9bf18f1a-3aa8-4ff5-95a6-e28ccd715941"], 1839 | Cell[CellGroupData[{ 1840 | Cell[23218, 702, 2471, 61, 135, "Input",ExpressionUUID->"cbd78080-dd56-4eed-917b-0913712b83eb"], 1841 | Cell[25692, 765, 2357, 52, 767, "Output",ExpressionUUID->"c83b7dd9-812d-4ed1-975d-0a437e0def88"] 1842 | }, Open ]], 1843 | Cell[28064, 820, 305, 7, 50, "Text",ExpressionUUID->"982fbac2-0444-4806-b90c-0147f3e125c1"], 1844 | Cell[28372, 829, 1128, 39, 203, "DisplayFormulaNumbered",ExpressionUUID->"b088b2d7-e9bc-4b53-a749-e7b3c93a3dda"], 1845 | Cell[29503, 870, 324, 10, 50, "Text",ExpressionUUID->"74622326-bf7a-47c4-8658-85b091169094"], 1846 | Cell[29830, 882, 1201, 44, 203, "DisplayFormulaNumbered",ExpressionUUID->"0b4d0c52-cbf2-4687-86db-4b43696ac384"], 1847 | Cell[31034, 928, 586, 14, 50, "Text",ExpressionUUID->"b866f5f3-8d2d-4ac8-b596-2e5d871ba899"], 1848 | Cell[CellGroupData[{ 1849 | Cell[31645, 946, 1102, 25, 74, "Input",ExpressionUUID->"324fcb3a-62cd-40ea-8b51-a5661155b587"], 1850 | Cell[32750, 973, 262, 5, 50, "Output",ExpressionUUID->"72083966-ad16-40ed-906a-0b32fda19dbc"] 1851 | }, Open ]], 1852 | Cell[33027, 981, 564, 15, 50, "Text",ExpressionUUID->"b34734be-ca68-4e4f-96e0-a098c533b3e5"], 1853 | Cell[33594, 998, 2613, 87, 204, "DisplayFormulaNumbered",ExpressionUUID->"deeadaaf-6df4-40a0-beb4-1ba9435eee18"], 1854 | Cell[CellGroupData[{ 1855 | Cell[36232, 1089, 929, 23, 74, "Input",ExpressionUUID->"aeb74690-ca2d-4648-a996-1eb1860fe6ac"], 1856 | Cell[37164, 1114, 1914, 40, 262, "Output",ExpressionUUID->"664a669a-27e8-489b-8679-0d7dbda1b40c"] 1857 | }, Open ]], 1858 | Cell[39093, 1157, 214, 5, 50, "Text",ExpressionUUID->"1247afad-fe42-4f0f-971f-ccfa094a6c23"], 1859 | Cell[CellGroupData[{ 1860 | Cell[39332, 1166, 2424, 61, 135, "Input",ExpressionUUID->"7fff56ac-3664-4d75-ad94-681d3feab7c5"], 1861 | Cell[41759, 1229, 2357, 52, 754, "Output",ExpressionUUID->"2fe809ca-0425-466c-b557-89220fd2ac41"] 1862 | }, Open ]], 1863 | Cell[44131, 1284, 472, 11, 83, "Text",ExpressionUUID->"d0348224-a8af-4533-9aef-7e21568e91f0"] 1864 | }, Open ]], 1865 | Cell[CellGroupData[{ 1866 | Cell[44640, 1300, 239, 4, 99, "Section",ExpressionUUID->"f043eb99-7f58-4e23-b4e4-eed166b4fbaf"], 1867 | Cell[CellGroupData[{ 1868 | Cell[44904, 1308, 1058, 24, 74, "Input",ExpressionUUID->"4991ece7-daac-43b8-92e6-598d1b51b9fa"], 1869 | Cell[45965, 1334, 770, 21, 82, "Output",ExpressionUUID->"10cc96f5-ea60-49e9-9c16-17da30e2a3fd"] 1870 | }, Open ]], 1871 | Cell[46750, 1358, 393, 10, 50, "Text",ExpressionUUID->"5526ed6a-bfce-470f-b85f-6eed436d44a9"], 1872 | Cell[CellGroupData[{ 1873 | Cell[47168, 1372, 768, 17, 77, "Input",ExpressionUUID->"ac8e91a1-6437-4cfa-a94b-d9b3447f856d"], 1874 | Cell[47939, 1391, 768, 21, 69, "Output",ExpressionUUID->"247ba9e8-7542-4712-a992-e364678c3c1d"] 1875 | }, Open ]], 1876 | Cell[48722, 1415, 630, 16, 50, "Text",ExpressionUUID->"8c9a5b71-4887-45d0-8740-75569fb37e83"], 1877 | Cell[49355, 1433, 475, 14, 104, "DisplayFormulaNumbered",ExpressionUUID->"5aba99b3-7ba1-493b-835a-0fc6346c8723"], 1878 | Cell[49833, 1449, 952, 25, 50, "Text",ExpressionUUID->"ea937e02-3b56-4786-bbd0-4b6d87b656a4"], 1879 | Cell[50788, 1476, 950, 29, 170, "DisplayFormulaNumbered",ExpressionUUID->"f18db181-8b31-4cca-834b-98caa4a2faf7"], 1880 | Cell[51741, 1507, 691, 17, 50, "Text",ExpressionUUID->"4da675c3-2c29-4d91-9fc9-1325c4a45aa0"], 1881 | Cell[CellGroupData[{ 1882 | Cell[52457, 1528, 1042, 23, 77, "Input",ExpressionUUID->"68ab044b-302e-4312-bde5-53ab17f47edf"], 1883 | Cell[53502, 1553, 1322, 40, 69, "Output",ExpressionUUID->"eae00a9c-888d-49d4-bbb4-68bb561ca41e"] 1884 | }, Open ]], 1885 | Cell[54839, 1596, 472, 13, 50, "Text",ExpressionUUID->"54ca97b7-e0f3-4ed1-8e65-193603ab60f7"], 1886 | Cell[CellGroupData[{ 1887 | Cell[55336, 1613, 852, 20, 77, "Input",ExpressionUUID->"db2e1247-6f85-49a8-b09c-dbb285317218"], 1888 | Cell[56191, 1635, 954, 26, 69, "Output",ExpressionUUID->"1823ebf5-aa1d-4f00-8e3e-268b4a1d10ec"] 1889 | }, Open ]], 1890 | Cell[57160, 1664, 331, 7, 50, "Text",ExpressionUUID->"6935125c-2c7c-45b3-b813-6774c2ade9ef"], 1891 | Cell[57494, 1673, 3288, 106, 223, "DisplayFormulaNumbered",ExpressionUUID->"5e3bc8b6-8d7d-48f8-b381-543f1e1e3a52"] 1892 | }, Open ]] 1893 | }, Open ]] 1894 | } 1895 | ] 1896 | *) 1897 | 1898 | --------------------------------------------------------------------------------