Page 896 - Advanced_Engineering_Mathematics o'neil
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876 Index
Fourier transforms (Continued) Frobenius solutions, 126–135
cosine function, 490–491, 586–587 Function space C[a, b], 181–186
defined, 471–472 distance between, 182
DFT approximation of, 501–504 dot product with weight function p
Dirac delta function δ(t) and, for, 182–183
481–482 linear dependence and independence,
discrete (DFT), 492–504 181–182
filtering and, 481–482 orthogonality of, 183–186
frequency differentiation, 479 scalar addition and multiplication,
frequency of signals ω, 471–472 181–182
frequency shifting, 476 Functions, 22–25, 43, 77–79, 81–82,
heat equation solutions using, 84–101,114–117, 121–122, 241,
627–628, 630 345–349, 367, 380–381, 429–440,
integrals, 479–481 452–456, 465–467, 483–485, 487,
inverse, 473–474, 494–495 511, 521–523, 533–560, 565–567,
linearity, 475, 480, 494 641, 709–711.
low-pass filters and, 487–488 See also Complex functions;
MAPLE commands for, 473–474 Eigenfunction expansions; Fourier
modulation, 477 series; Special functions
operational rule for, 477–478 analytic, 121–122
pair, 473 availability f (t),99
scaling, 476 Bessel, 114–117, 533–560
Shannon sampling theorem and, Cesàro filter Z(t), 462–463
485–486 Cesàro sum σ(t), 461
sine function, 490–491, 586–587, 630 characteristic, 487
symmetry, 477 convolution theorem for, 96–101
time reversal, 476 coordinate, 367
time shifting, 475–476 Dirac delta δ(t),102–106
wave (motion) equation solution eigenfunction expansions and, 511,
using, 582–584, 586–587 521–523, 533–534
windowed, 483–485 even, 436–438
Free radiation, 612 filter Z, 461–462
Free variables, 214–215 forcing f , 43, 77–79
Frequency ω, 460, 471–472, 476, Fourier integrals and, 465–467
479–480, 604 Fourier series of, 429–440, 452–456
convolution, 480 fundamental period of, 452–454
differentiation, 479 gamma (x), 533–534
Fourier transforms, 476, 479–480 generating, 521–523, 548–549
normal modes, 604 harmonic, 454–455, 641, 709–711
shifting, 476 Heaviside H, 86–95
signals ω, 471–472 jump discontinuities, 81, 86–87
spectrum, 460 Laplace transforms of, 77–79
vibration, 604 Legendre polynomials, 521–523
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