Page 15 - Advanced engineering mathematics

Contents ix

14.3.3 The Shannon Sampling Theorem 485
14.3.4 Low-Pass and Bandpass Filters 487
14.4 Fourier Cosine and Sine Transforms 490
14.5 The Discrete Fourier Transform 492
14.5.1 Linearity and Periodicity of the DFT 494
14.5.2 The Inverse N-Point DFT 494
14.5.3 DFT Approximation of Fourier Coefficients 495
14.6 Sampled Fourier Series 498
14.7 DFT Approximation of the Fourier Transform 501

CHAPTER 15 Special Functions and Eigenfunction Expansions 505
15.1 Eigenfunction Expansions 505
15.1.1 Bessel’s Inequality and Parseval’s Theorem 515
15.2 Legendre Polynomials 518
15.2.1 A Generating Function for Legendre Polynomials 521
15.2.2 A Recurrence Relation for Legendre Polynomials 523
15.2.3 Fourier-Legendre Expansions 525
15.2.4 Zeros of Legendre Polynomials 528
15.2.5 Distribution of Charged Particles 530
15.2.6 Some Additional Results 532
15.3 Bessel Functions 533
15.3.1 The Gamma Function 533
15.3.2 Bessel Functions of the First Kind 534
15.3.3 Bessel Functions of the Second Kind 538
15.3.4 Displacement of a Hanging Chain 540
15.3.5 Critical Length of a Rod 542
15.3.6 Modified Bessel Functions 543
15.3.7 Alternating Current and the Skin Effect 546
15.3.8 A Generating Function for J ν (x) 548
15.3.9 Recurrence Relations 549
15.3.10 Zeros of Bessel Functions 550
15.3.11 Fourier-Bessel Expansions 552
15.3.12 Bessel’s Integrals and the Kepler Problem 556

PART 5 Partial Differential Equations 563

CHAPTER 16 The Wave Equation 565
16.1 Derivation of the Wave Equation 565
16.2 Wave Motion on an Interval 567
16.2.1 Zero Initial Velocity 568
16.2.2 Zero Initial Displacement 570
16.2.3 Nonzero Initial Displacement and Velocity 572
16.2.4 Influence of Constants and Initial Conditions 573
16.2.5 Wave Motion with a Forcing Term 575

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October 15, 2010 17:43 THM/NEIL Page-ix 27410_00_fm_pi-xiv