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Index

Abel's integral equation, 107 abscissa of convergence, 28, 331 adjoint

boundary conditions, 165 problem, 165

advanced potential, 188 Airy functions, 325 albedo problem, 295 analytic, 4

continuation, 9-12 function, 4 functionals, 153-156

anomalous system, 71 asymptotic expansion, 34, 50-53,

197-200, 215-222 asymptotically equal, 34

Barnes, 205 Bernoulli's equation, 133 Bessel functions, 310-318

of the first kind, 66, 115, 312 Fourier transform of, 115 integral representations, 310-314,

318-320 integrals involving, 324

of the second and third kind, 314-318

Bessel's equation, 65, 310 Bessel's integral, 115, 184, 313 beta function, 23 block diagrams, 61 branch cut, 5 branch point

definition, 4 integrals around, 15 inversions involving, 49-52, 100,

116, 174, 256, 298, 307 Bromwich contour, 43, 353

Carleman, 265 Case and Zweifel, 295 Cauchy integral formula, 9 Cauchy integrals, 283-288 Cauchy's theorems, 8 Cauchy-Riemann relations, 3 causality, 121 Chebyshev polynomials, 335-338 Clenshaw's algorithm, 343 complementary error function, 88,

208

364 Index

continued fractions, 350 continuity oflinear functionals, 147 contour, 2

integration, 6 controllability, 75 convergence

of generalized functions, 150 of test functions, 147

convolution equations, 97-104 convolutions, 32, 58, 87, 117, 122,

182, 201, 218 cosine transform, 111 Coulomb gauge, 187 Cramer's rule, 71 cylinder functions, 227, 315

D'Alembert's method, 194 delta function, 143, 147 diagonal Pade approximation, 350 diffraction problems, 185-187, 265-

272, 297-301 diffusion problems, 85-90 Dirac's delta function, 143 direct correlation function, 105 Dirichlet conditions, 41 Dirichlet integrals, 41 discontinuity theorem, 288 distributions, 154 double Laplace transforms, 192-

194 dual integral equations, 236-239

eigenfunction expansion, 253 electric circuit problems, 68-70 electron gas, 220 electrostatic problems, 129-131, 191,

209, 233, 236 entire function, 18 epsilon algorithm, 347 Erdelyi-Kober operators, 239-242 Euler's constant, 12, 199, 221 exponential integral, 12

factorial function, 19-22 asymptotic expansion, 216

functional relations, 20 Hankel's integral representation,

22 fast Fourier transform, 341 feedback loop, 63 Fourier integrals, ascending expan­

sions for, 221 Fourier series, 229, 252, 258, 343 Fourier transform

application to partial differen-tial equations, 129-140

definition, 111 of generalized functions, 155 inverse of, 112 properties of, 116-118 relation to Green's functions, 254 relation to Hankel transform, 229 relation to Laplace transform,

111 sine and cosine transforms, 112 of test functions, 145 in two or more variables, 181-

189 fractional integration, 239 Fraunhofer diffraction, 186 Fresnel diffraction, 186 functional, 144

analytic, 153-156 continuous, 147 linear, 146 regular, 147 singular, 147

generalized functions, 143-157, 167, 188, 276 convergence of, 150 definition, 146 differentiation of, 149 on finite interval, 148 Fourier transforms of, 155 properties of, 147-151 regular, 147 sequences of, 150 singular, 147

Green's functions, 163-166

for adjoint, 165 as generalized functions, 167 for Helmholtz's equation, 173 integral transforms generated by,

249 one-dimensional, 163 for Poisson's equation, 169 symmetry of, 171

Green's theorem, 7

Hankel functions, 173, 228, 272, 317-320

Hankel transform, 227 application to boundary-value

problems, 232 connection with Fourier trans-

form, 229 definition, 227 inverse of, 227 properties of, 230, 231 relation to Green's functions, 255

Hankel's loop integral, 17 harmonic function, 9, 133, 246, 258 heat conduction, 86-90 heat diffusion kernel, 87 Heaviside

distortionless line, 93 expansion theorem, 48 series expansion, 53 step function, 28

Helmholtz's equation, 173-176, 266 elementary solution, 173 Green's function for, 176

Hermite equation, 305 Hermite functions, 307-310

asymptotic forms, 205-207 Hermite polynomials, 305-307 Holder condition, 285 Hopf,265 hydrodynamic equations, 132

images, 172 impedance, 93 influence function, 122 integral equations, 97-107, 274, 292

classification, 97 dual, 236-239

integrals Fourier, 221

Index 365

involving a parameter, 218 integro-differential equations, 274 inverse Fourier transform, 112

sine and cosine transform, 113 inverse Laplace transform, 39

asymptotic forms of, 50-54 involving a branch point, 49 of meromorphic functions, 47 numerical evaluation of, 327-355 of rational functions, 44 Taylor series of, 46

Jacobi polynomials, 335

Kirchhoff, 185 Kontorovich-Lebedev transform,

256-262 relation to Mellin transform, 258

Kramers-Kronig relations, 121

Lagrangian interpolation, 333 Laguerre polynomials, 210, 334,

338 Laplace transform

application to ordinary differ­ential equations, 59

application to partial differen­tial equations, 85-93

application to simultaneous dif-ferential equations, 67

asymptotic properties, 33, 52 definition, 27 differential equations with poly-

nomial coefficients, 65 double, 192-194 inverse of, 39 inversion theorem, 42 properties of, 28-32 relation to Fourier transform,

111 Watson's lemma, 35, 50

366 Index

Laplace's equation, 129, 190, 233 Laplace's method, 66, 303-321 Laurent expansion, 13 Lienard-Wiechert potential, 189 linear control theory, 72-82

controllability, 75 equivalent systems, 77 minimal realization, 79 observability, 78 realization, 79

linear functionals, 146 linear transport theory, 291-297 Liouville's theorem, 18 Lommel's integral, 228 loop integrals, 15, 49-53

Macdonald's function, 321 MacRobert, 227 matrix exponential, 73 Maxwell's equations, 187 Mellin transform, 195

application to differential equa­tions, 205

application to potential prob­lems, 202

in asymptotics, 197-200, 215-222

definition, 195 inverse of, 196 properties of, 200-203 relation to Fourier transform,

195 relation to Green's functions, 255 in summation, 211-216

meromorphic functions, 14, 47 inverse Laplace transform of, 47

method of images, 172 Milne's equation, 274, 284 minimal realization of transfer func-

tion, 79 Mittag-Leffler theorem, 281 Mobius transformation, 331 modified Bessel functions, 320

Newton's law of cooling, 89

Newton's second law, 132 normal system, 71 numerical inversion of Laplace trans­

forms, 327-355 collocation methods, 333 Fourier series methods, 343 Gaver-Stehfest method, 329 Korrectur method, 346 Lyness and Giunta's method,

340 method of de Hoog, Knight, and

Stokes, 349 Talbot's method, 352 Weeks's methods, 339

observability, 78 ordinary differential equations

Green's functions for, 163-169 Laplace transform methods for,

57-77, 79-82 Laplace's method for, 303-321 stability of solutions, 60

Pade approximation, 349 pair distribution function, 104 Parseval relations, 118, 121, 231 partial differential equations

Fourier transform methods for, 129-140

Laplace transform methods for, 85-93

partial fractions, 45 Percus-Yevick equation, 104 Plemelj formulae, 286-289 Poisson integral representation, 225,

319 Poisson summation formula, 128,

153 Poisson's equation, 169 pole, 13 polynomial interpolation, 333 potential problems, 129-132, 187-

189, 202, 233-237 power series, 10

asymptotic behavior of, 215-217

principal value integral, 122, 152, 286

quotient-difference algorithm, 351

radiation condition, 139, 184, 188 rational functions

inverse Laplace transform of, 44 realization of transfer functions,

79 recurrence relations, 336, 342, 351 regular generalized functions, 147 regularization, 328 residue theory, 13-15 resolvent kernel, 98 retarded potential, 187 Riemann zeta function, 23-26

asymptotic forms, 26 functional relation, 25 in summation, 212

Riemann-Hilbert problem, 289-291

self-adjoint, 166, 176, 249 shrinking a contour, 15 simple pole, 13 sine transform, 111 singular generalized functions, 147 singular point, 4 singularity, 4 Sommerfeld diffraction problem,

265-272 Sonine's integrals, 243

Index 367

spectral analysis, 119-121 stability of solutions, 60 Stirling's series, 216 stretched string, 90--93 Sturm-Liouville problem, 251 symmetry of Green's functions, 171

Taylor series of inverse Laplace transform, 46

test functions, 144-146 Titchmarsh, 237, 251 transfer functions, 61 transmission line, 92, 93 trapezoidal rule, 340, 344, 353 two-point boundary-value problem,

164

ultradistributions, 154

variation of parameters, 164

Watson's lemma, 33-36 for loop integrals, 50--53

wave equation, 85, 90, 173, 185, 259, 265, 297

wave propagation, 90 Weber functions, 317 Weber's integral, 234 Wiener, 265 Wiener-Hopf technique, 265

zeta function, 23

Texts in Applied Mathematics

(continued from page ii)

31. Bremaud: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. 32. Durran: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics.

33. Thomas: Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations.

34. Chicone: Ordinary Differential Equations with Applications. 35. Kevorkian: Partial Differential Equations: Analytical Solution Techniques, 2nd ed.

36. DulierudiPaganini: A Course in Robust Control Theory: A Convex Approach. 37. QuarteroniiSacco/Saleri: Numerical Mathematics. 38. Gallier: Geometric Methods and Applications: For Computer Science and

Engineering. 39. Atkinson/Han: Theoretical Numerical Analysis: A Functional Analysis Framework. 40. Brauer/Castillo-Chavez: Mathematical Models in Population Biology and

Epidemiology. 41. Davies: Integral Transforms and Their Applications, 3rd ed.