Index

c, 400 recursion formula, 234 classical orthogonal polynomials, 215, 219 Ørstead, Christian, 397 second kind, 233 coefficient matrix, 136 matrix:symmetric, 160 zeros, 235 cofactors, 126 Bessel’s inequality, 253 commutation operation, 160 adjoint operator, 243 Bessel, Friedrich Wilhelm, 232 complete basis, 254 Ampère,André-Marie, 397 big-Oh, 34, 491, 493 completely antisymmetric symbol, 378 amplitude, 55 binomial coefficients, 30, 487 complex differentiation, 277 analog signal, 343 binomial expansion, 30, 487 complex functions, 271 analytic function, 288 Bose-Einstein integrals, 355 multivalued, 272 angular frequency, 55, 332 boundary value problem, 167 natual logarithm, 273 anticommutative, 377 box function, 336 real and imaginary parts, 272 arclength, 385 branch cut, 273 complex numbers, 267 Argand diagram, 267 branch point, 307 addition, 268 associated Laguerre polynomials, 462 Bromwich integral, 368 Cartesian form, 268 Associated Legendre Functions, 451 Bromwich, Thomas John l’Anson, 353 complex conjugate, 269 associated Legendre functions, 227 Buckingham Π Theorem, 39 imaginary part, 267 atomic bomb example, 41 Buckingham, edgar, 39 modulus, 267 autonomous, 87 multiplication, 268 cakes nth root, 269 BAC-CAB rule, 382 cylindrical, 444 nth roots of unity, 270 back of the envelope calculations, 40 rectangular, 442 polar form, 268 beats, 72 capacitor, 61 quotient, 268 Beeblebrox, Zaphod, 412 Cardano, Girolamo, 267 real part, 267 Bernoulli numbers, 355 Carlo, Giulio, 105 complex plane, 267 Bernoulli, Daniel, 166, 267 Cauchy Integral Formula, 290 conic equation, 151 Bernoulli, Jakob, 267 Cauchy principal value integral, 302 conics, 151 Bernoulli, Jakob II, 267 Cauchy’s Theorem, 284 ellipse, 152 Bernoulli, Johann, 267 Cauchy, Augustin-Louis, 278 general equation, 151, 154 Bernoulli, Johann II, 267 Cauchy-Euler equations, 72 hyperbola, 155 Bernoulli, Johann III, 267 nonhomogeneous, 75 connected set, 280 Bernoulli, Nicolaus II, 267 Cauchy-Riemann equations, 278 constant coefficient equations, 57 Bernoulli, Nikolaus I, 267 center, 95 constant coefficient equations:complex Bessel function, 428 Chain Rule, 18 roots, 59 Bessel functions, 232 characteristic equation, 58, 74, 135 constant coefficient equations:repeated first kind, 233 charge density, 398 roots, 58 Fourier-Bessel series, 235 chemical kinetics, 149 constant coefficient systems, 137 generating function, 235 chili problem, 163, 164 continuity equation, 399 identities, 234 circle of convergence, 288 contour deformation, 285 orthogonality, 234 circular membrane, 424 contour integral, 303 502 mathematical physics

convergence nonhomogeneous, 67 Euler angles, 121 absolute, 474 second order, 56 Euler’s Formula, 486 conditional, 474 separable, 49 Euler’s method, 80 pointwise, 475 differential equations, 356 Euler, Leonhard, 105, 166, 267 real sequences, 467 coupled, 130 even functions, 188 uniform, 476, 478 differential operator, 56 exponential of a matrix, 136 convergence in the mean, 253 Differentiation Under Integral, 24 convergence tests, 470 dimensional analysis, 39 Faraday’s law, 398 Comparison test, 471 Dirac delta function, 329 Faraday, Michael, 397 Integral test, 471 Laplace transform, 361 Feynman’s trick, 24 Limit comparison test, 471 sifting property, 330 Feynman, Richard, 15, 24, 354 M-test, 478 Dirac, Paul Adrien Maurice, 114, 329 Fibonacci, Leonardo Pisano, 466 nth term divergence test, 470 direction field, 92 field, 112 Ratio test, 472 directional derivative, 388 filtering, 343 Root test, 473 Dirichlet boundary conditions, 238 finite wave train, 339, 344 convolution Dirichlet kernel, 205 flux, 393 Fourier transform, 339 dispersion relation, 318 Fourier analysis, 180 Laplace transform, 364 displacement current, 399 Fourier coefficients, 180 convolution theorem distribution, 329 Fourier series, 180 Fourier transform, 339 divergence, 390 complex exponential, 325 Laplace transform, 363 divergenceless, 391 Maple code, 197 coordinates domain, 281 representation on [0, 2π], 181 cylindrical, 407 domain coloring, 273 Fourier transform, 326 polar, 117 dot product, 375 convolution, 334 cosine series, 194 double factorial, 224 properties, 332 Coulomb gauge, 402 double tank problem, 148 shifitng properties, 333 coupled systems, 87 drag force, 53 Fourier, Joseph, 166 Cramer’s rule, 90, 128 Dyson, Freeman, 3 Fourier-Bessel series, 232 cross product, 375 Fourier-Legendre series, 219, 228 cube roots of unity, 271 eigenfunction expansion, 249 Fredholm alternative, 254 curl, 390 eigenvalue, 132 free fall, 45 current, 61 eigenvalue problem, 132, 135 frequency, 55, 332 current density, 398 generalized, 159 function space, 212 curvilinear coordinates, 404 eigenvector, 132 functions cutoff frequency, 343 Einstein summation convention, 380 exponential, 12 electric dipole, 413 hyperbolic, 16 d’Alembert, Jean le Rond, 166 electric field, 398 logarithmic, 12 d’Alembertian, 404 electric permittivity, 401 polynomial, 12 damped harmonic motion, 96 electric potential, 401 rational, 12 de Moivre’s Formula, 486 electromagnetic waves, 373, 400 trigonometric, 13 de Moivre, Abraham, 486 ellipse, 151 functions:inverse trigonometric functions, de Vries, Gustav, 318 elliptic integral, 104, 105 15 derivatives, 11 entire function, 288 Fundamental Theorem of Calculus, 19, 46 table, 18 epidemic model, 150 fundamental theorem of calculus, 391 determinant, 125 equilibrium, 89 diagonalization, 131, 155 center, 95 Gamma function, 230, 352 difference equation, 351 degenerate node, 97 gate function, 336, 344 differential equation focus, 96 Gauß, Carl Friedrich, 469 autonomous, 49 node, 95 Gauss’ law, 398 first order, 49, 50 saddle, 94 Gaussian function, 334, 342 linear, 49, 56 source, 95 Gaussian integral, 335 index 503

Gegenbauer polynomials, 219 Pythagorean, 14 differential equations, 356 generalized function, 329 sum and difference, 14 inverse, 367 geometric series, 27, 288, 292 tangent, 14 properties, 352 Gibbs phenomenon, 198, 204 implicit solution, 50 series summation, 353 Gibbs, Josiah Willard, 204 impulse function, 344 Laplace’s equation, 433, 448 Goeppert-Mayer, Maria, 45 unit impulse, 362 Polar coordinates, 438 gradient, 388 impulse response, 344 Rectangular coordinates, 434 Gram-Schmidt Orthogonalization, 216 inductor, 61 Laplace, Pierre-Simon, 346 gravitational potential, 223 infinite dimensional, 212 Laplace,Pierre-Simon, 448 Green’s identity, 246 initial value problem, 49, 357 Laplacian, 169, 373 Green’s Theorem in the Plane, 284 inner product, 212 polar coordinates, 425 Green,George, 284 inner product space, 213 Laurent series, 293 Gregory, James, 479 integral transforms, 347 singular part, 294 group velocity, 322 integrals, 11 Laurent, Pierre Alphonse, 293 integration limits, 20 law of cosines, 375 Hamilton, William Rowan, 399 simple substitution, 19 Law of Gravitation, 413 harmonic conjugate, 279 table, 19 LC circuit, 64 harmonic functions, 278, 449 trigonometric, 26 least squares approximation, 251 harmonics, 166 integrating factor, 50 Legendre polynomials, 219, 451 meridional, 456 integration by parts, 21, 190 generating function, 223 sectoral, 456 integration by recursion, 351 leading coefficient, 222 spherical, 456 interval of convergence, 483 normalization, 228 surface, 456 inverse Fourier transform, 326 recurrence relation, 220 tesseral, 456 inverse Laplace transform, 357, 367 recursion formula, 220 zonal, 456 irrotational, 391 Rodrigues formula, 220 Heart of Gold, 412 Legendre, Adrien-Marie, 218, 230 heat equation, 170 Jacobi polynomials, 219 Leibniz’s Theorem, 474 1D, 170, 199 Jacobian determinant, 407 Leibniz, Gottfried Wilhelm, 474 Heaviside function, 229, 359 Jordan canonical form, 158 Levi-Civita symbol, 378 step function, 359 Jordan’s lemma, 304, 311 limit theorems, 467 Heaviside, Oliver, 229 Julia set, 274 line of equilibria, 99 Heisenberg, Werner, 337 linear operator, 56 Helmholtz equation, 418 Kepler, Johannes, 166, 232 matrix representation, 122 Helmholtz’s Theorem, 401 kernel, 317 linear transformation, 121 Helmholtz, Ludwig, 401 kettle drum, 424 linearity, 56, 332 Hermite polynomials, 219, 259 Kirchoff’s rules, 61, 145 logarithm Hitchhiker’s Guide, 412 Kortweg, Diederik, 318 multi-valued, 273 Holmes, Sherlock, 465 Kortweg-deVries equation, 317 principal value, 273 holomorphic function, 278, 288 Kronecker delta, 114, 214, 381 LRC circuits, 61, 145 homogeneous, 87 Kronecker, Leopold, 114 Hooke’s law, 85 Maclaurin series, 136 hydrogen atom, 459 L’Hopital’s Rule, 468 Maclaurin, Colin, 479 hyperbolic cosine, 16 Lagrange’s identity, 246 magnetic field, 398 hyperbolic function identities, 17 Lagrange, Joseph-Louis, 232 magnetic permeability, 401 hyperbolic sine, 16 Laguerre polynomials, 219, 462 magnetic potential, 401 hyperbolic tangent, 16 Laguerre, Edmond, 462 Mandelbrot set, 274 Laplace operator, 391 mass-spring system, 54 identities Laplace Transform coupled, 86 double angle, 14 transform pairs, 348 MATLAB code, 274 half angle, 14 Laplace transform, 346 matrices, 119 product, 14 convolution, 363 matrix 504 mathematical physics

determinant, 125 orbit, 91 potential function, 401 Hermitian, 132 ordinary differential equation, 49 predator-prey model, 147 identity, 121 orthogonal functions, 183 principal axes, 152 inverse, 120 orthonormal, 183 principal vectors, 154 multiplication, 119 oscillations product solutions, 173 real symmetric, 132 coupled, 85 similar, 156 damped, 66 quaternions, 397 singular, 126 forced, 71 trace, 127 underdamped, 67 radius of convergence, 288, 483 transpose, 120 overdamped, 97 Rayleigh quotient, 247 matrix system, 136 RC circuits, 62 matrix:antisymmetrc, 160 p-test, 472 rectangular membrane, 419 matrix:inverse, 126 parallelepiped, 382 relative error, 103 Maxwell’s equations, 397 parametrization, 281 residue, 296 Maxwell, James Clerk, 397 Parseval’s equality, 254, 345 Residue Theorem, 298 Maxwell-Ampère Law, 398 Parseval, Marc-Antoine, 345 resistor, 61 mean square deviation, 252 partial differential equation, 169 resonance, 71 mechanical energy, 104 partial fraction decomposition, 53, 357 Riemann surface, 273, 307 membrane partial fractions, 293 Riemann zeta function, 354 annular, 432 partial sum, 469 Riemann, Georg Friedrich Bernhard, 278 circlur, 424 particle wave function, 320 Riemann-Lebesque Lemma, 254 rectangular, 419 particular solution, 49 Romeo and Juliet, 146 meromorphic function, 288 partition function, 29 rotation matrix, 123 Method of Undetermined Coefficients, 60, Pascal’s triangle, 31, 488 rotation of conics, 151 68 Pascal, Blaise, 31, 488 rotations, 118 modified, 70 path independence, 282 Russell, Bertrand, 11 method of undetermined coefficients, 57 path integral, 280 Russell, J. Scott, 318 mixture problems, 147 Pauli matrices, 160 momentum, 321 pendulum, 54 saddle point, 94 Morera’s Theorem, 288 nonlinear, 102 scalar field, 387 Morera, Giacinto , 287 period, 40 scalar product, 183 multivalued functions, 270, 307 period, 55, 102, 178 scale factors, 405 integration, 309 periodic boundary condition, 239 scheme periodic boundary conditions, 427 algebraic system, 323 Neumann boundary conditions, 238 periodic extension, 180, 191 Laplace transform, 347 Neumann function, 428 periodic function, 15, 178 Schrödinger equation, 322 Neumann functions, 233 permutation symbol, 378 Schrödiger equation , 320 Newton’s Law of Cooling, 464 phase, 179 Schrödinger equation, 458 Newton’s second law, 46, 55 phase portrait, 91 self-adjoint, 245 Newton, Isaac, 474 phase shift, 179 separation of variables, 171 nodal curves, 421 phase velocity, 322 sequence nodal lines, 421 Plancherel’s formula, 345 Fibonacci, 466 nonautonomous, 87 Plancherel, Michel, 345 functions, 475 norm of functions, 215 Poisson equation, 401 real numbers, 466 normal, 283 Poisson Integral Formula, 441 recursive, 466 normalization, 183 Poisson kernel, 441 series normalize basis functions, 215 polar coordinates, 100, 117 alternating series, 474 numerical solutions, 79 derivatives, 100 binomial series, 33, 490 poles, 294 Fourier series, 180 odd functions, 188 potential drops, 61 functions, 477 open set, 280 potential energy, 104 geometric series, 27 index 505

harmonic series, 472 step function, 359 vector field, 387, 389 Laurent series, 291 Stone-Weierstraß Theorem, 216 vector functions, 383 Maclaurin series, 481 Sturm-Liouville operator, 238 vector identities, 395 p-series, 472 Sturm-Liouville problem, 235, 237, 260, vector projection, 217 power series, 479 261 vector space, 56, 111 re-indexed, 292 sum of an infinite series, 470 function space, 212 real numbers, 469 systems inner product space, 213 summation by transforms, 353 constant coefficient, 87 vector spaces, 111 Taylor series, 480 finite dimensional, 111 telescoping, 29, 496 Tabular Method, 23 vectors similarity matrix, 156 Taylor polynomials, 480 components, 113 simple closed contour, 283 Taylor’s series, 288 length, 113 simple harmonic motion, 54, 85, 97, 361 Taylor, Brook, 166, 479 linearly independent, 113 simple pendulum, 55 Tchebychef polynomials, 219 orthogonal basis, 114 sinc function, 336 terminal velocity, 52 orthonormal basis, 114 sine series, 194 time constant, 63 scalar product, 114 double, 423 tones, 177 standard basis, 113 singularity, 294 torque, 376 vibrating string, 170 double pole, 295 trajectory, 91 essential, 294 transformation Wallis, John, 166 poles, 294 similarity, 131 Watson, G.N., 232 removable, 294 transformations wave equation, 166, 170 simple pole, 295 linear, 117 1D, 174, 201 slope field, 91 rotation, 118 electromagnetic, 400 small angle approximation, 56 Trigonometric identities, 182 rectangular, 419 solenoidal, 391 trigonometric identities, 14 wave packet, 321 solitons, 16 triple cross product, 382 wave speed, 318 solutions triple scalar product, 381 wavelength, 318 equilibrium, 89 wavenumber, 318, 332 space of square integrable functions, 253 uncertainty principle, 337 Weierstraß, Karl Theodor Wilhelm, 478 spacecurve, 384 underdamped, 97 Weierstraß M-Test, 478 special limits, 467 units Weierstraß substitution, 301 speed of light, 400 basic, 38 Wilbraham, Henry, 204 spherical harmonics, 449, 453 unstable node, 95 Wilkinson, Rebecca, 441 spherical symmetry, 448 windowing, 344 spiral, 96, 100 variation of parameters, 57, 76 work, 375 square wave, 360 vector, 111 Wronskian, 57