.

0 (hodnocen0 x )

(1) Půjčeno:1x 

BK

Příručka

3rd ed.

New York : Springer, c2002

xvii, 367 s.

ISBN 0-387-95314-0 (váz.)

Texts in applied mathematics ; 41

Obsahuje předmluvy, rejstřík

Bibliografie: s. 357-362

Transformace integrální - sborníky

000040520

Preface to the Third Edition vii // Preface to the Second Edition ix // Preface to the First Edition xi // 1 Functions of a Complex Variable 1 // 1.1 Analytic Functions 1 // 1.2 Contour Integration 6 // 1.3 Analytic Continuation 9 // 1.4 Residue Theory 13 // 1.5 Loop Integrals 15 // 1.6 Liouville’s Theorem 18 // 1.7 The Factorial Function 19 // 1.8 Riemann’s Zeta Function 23 // 2 The Laplace Transform 27 // 2.1 The Laplace Integral 27 // 2.2 Important Properties 28 // 2.3 Simple Applications 32 // 2.4 Asymptotic Properties: Watson’s Lemma 33 // Problems 36 // 3 The Inversion Integral 39 // 3.1 The Riemann-Lebesgue Lemma 39 // xiv // Contents // 3.2 Dirichlet Integrals 41 // 3.3 The Inversion 42 // 3.4 Inversion of Rational Functions 44 // 3.5 Taylor Series Expansion 46 // 3.6 Inversion of Meromorphic Functions 47 // 3.7 Inversions Involving a Branch Point 49 // 3.8 Watson’s Lemma for Loop Integrals 50 // 3.9 Asymptotic Forms for Large t 52 // 3.10 Heaviside Series Expansion 53 // Problems 54 // 4 Ordinary Differential Equations 57 // 4.1 Elementary Examples 57 // 4.2 Higher-Order Equations 59 // 4.3 Transfer Functions and Block Diagrams 61 // 4.4 Equations with Polynomial Coefficients 65 // 4.5 Simultaneous Differential Equations 67 // 4.6 Linear Control Theory 72 // 4.7 Realization of Transfer Functions 79 // Problems 82 // 5 Partial Differential Equations 1 85 // 5.1 Heat Diffusion: Semi-Infinite Region 86 // 5.2 Finite Thickness 89 // 5.3 Wave Propagation 90 // 5.4 Transmission Line 92 // Problems 94 // Integral Equations // 6.1 Convolution Equations of Volterra Type // 6.2 Convolution Equations over an Infinite Range // 6.3 The Percus-Yevick Equation // Problems // 97 // 97 // 101 // 104 // 107 // 7 The Fourier Transform 111 // 7.1 Exponential, Sine, and Cosine Transforms Ill // 7.2 Important Properties 116 // 7.3 Spectral Analysis 119 //

7.4 Kramers-Kronig Relations 121 // Problems 123 // 8 Partial Differential Equations II 129 // 8.1 Potential Problems 129 // 8.2 Water Waves: Basic Equations 132 // 8.3 Waves Generated by a Surface Displacement 135 // 8.4 Waves Generated by a Periodic Disturbance 137 // Problems 140 // 9 Generalized Functions 143 // 9.1 The Delta Function 143 // 9.2 Test Functions and Generalized Functions 144 // 9.3 Elementary Properties 148 // 9.4 Analytic Functionals 153 // 9.5 Fourier Transforms of Generalized Functions 155 // Problems 157 // 10 Green’s Functions 163 // 10.1 One-Dimensional Green’s Functions 163 // 10.2 Green’s Functions as Generalized Functions 167 // 10.3 Poisson’s Equation in Two Dimensions 169 // 10.4 Helmholtz’s Equation in Two Dimensions 173 // Problems 176 // 11 Transforms in Several Variables 181 // 11.1 Basic Notation and Results 181 // 11.2 Diffraction of Scalar Waves 185 // 11.3 Retarded Potentials of Electromagnetism 187 // Problems 189 // 12 The Mellin Transform 195 // 12.1 Definitions 195 // 12.2 Simple Examples 196 // 12.3 Elementary Properties 200 // 12.4 Potential Problems in Wedge-Shaped Regions 202 // 12.5 Transforms Involving Polar Coordinates 203 // 12.6 Hermite Functions 205 // Problems 207 // 13 Application to Sums and Integrals 211 // 13.1 Mellin Summation Formula 211 // 13.2 A Problem of Ramanujin 213 // 13.3 Asymptotic Behavior of Power Series 215 // 13.4 Integrals Involving a Parameter 218 // 13.5 Ascending Expansions for Fourier Integrals 221 // Problems 223 // 14 Hankel Transforms 227 // 14.1 The Hankel Transform Pair 227 // 14.2 Elementary Properties 230 // 14.3 Some Examples 231 // 14.4 Boundary-Value Problems 232 // xvi Contents // 14.5 Weber’s Integral 234 // 14.6 The Electrified Disc 236 // 14.7 Dual Integral Equations of Titchmarsh Type 237 // 14.8 Erdelyi-Kober Operators 239 // Problems 242 //

15 Integral Transforms Generated by Green’s Functions 249 // 15.1 The Basic Formula 249 // 15.2 Finite Intervals 251 // 15.3 Some Singular Problems 253 // 15.4 Kontorovich-Lebedev Transform 256 // 15.5 Boundary-Value Problems in a Wedge 258 // 15.6 Diffraction of a Pulse by a Two-Dimensional Half-Plane 259 // Problems 262 // 16 The Wiener-Hopf Technique 265 // 16.1 The Sommerfeld Diffraction Problem 265 // 16.2 Wiener-Hopf Procedure: Half-Plane Problems 273 // 16.3 Integral and Integro-Differential Equations 274 // Problems 278 // 17 Methods Based on Cauchy Integrals 283 // 17.1 Wiener-Hopf Decomposition by Contour Integration 283 // 17.2 Cauchy Integrals 285 // 17.3 The Riemann-Hilbert Problem 289 // 17.4 Problems in Linear Transport Theory 291 // 17.5 The Albedo Problem 295 // 17.6 A Diffraction Problem 297 // Problems 302 // 18 Laplace’s Method for Ordinary Differential Equations 303 // 18.1 Laplace’s Method 303 // 18.2 Hermite Polynomials 305 // 18.3 Hermite Functions 307 // 18.4 Bessel Functions: Integral Representations 310 // 18.5 Bessel Functions of the First Kind 312 // 18.6 Functions of the Second and Third Kind 314 // 18.7 Poisson and Related Representations 319 // 18.8 Modified Bessel Functions 320 // Problems 321 // 19 Numerical Inversion of Laplace Transforms 327 // 19.1 General Considerations 327 // 19.2 Gaver-Stehfest Method 329 // 19.3 Mobius Transformation 331 // Contents xvii // 19.4 Use of Chebyshev Polynomials 335 // 19.5 Use of Laguerre Polynomials 338 // 19.6 Representation by Fourier Series 343 // 19.7 Quotient-Difference Algorithm 349 // 19.8 Talbot’s Method 352 // Bibliography // Index