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Davies B. — Integral Transforms and Their Applications
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Название: Integral Transforms and Their ApplicationsАвтор: Davies B. Аннотация: This book is an introduction to and a reference for the applications of integral problems to a wide range of common mathematical problems. Emphasis is placed on the development of techniques and on the connection between properties of transforms and the kinds of problems for which they provide tools. Over 400 problems accompany the text, illustrating areas of applications. This third edition has been substantially updated, extended, and reorganized. Graduate students and researchers working in mathematics and physics will find this book useful.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: Third EditionГод издания: 2002Количество страниц: 392Добавлена в каталог: 25.06.2008Операции: Положить на полку |
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Abel's integral equation 107 Abscissa of convergence 28 331 Adjoint boundary conditions 165 Adjoint problem 165 Advanced potential 188 Airy functions 325 Albedo problem 295 Analytic 4 Analytic continuation 9-12 Analytic function 4 Analytic functionals 153-156 Anomalous system 71 Asymptotic behavior of 215-217 Asymptotic expansion 34 50-53 197-200 215-222 Asymptotically equal 34 Barnes 205 Bernoulli's equation 133 Bessel functions 310-318 Bessel functions, Fourier transform of 115 Bessel functions, integral representations 310-314 318-320 Bessel functions, integrals involving 324 Bessel functions, of the first kind 66 115 312 Bessel functions, 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 Branch point, integrals around 15 Branch point, inversions involving 49-52 100 116 174 256 298 307 Bromwich contour 43 353 Carleman 265 Case and Zweifel 295 Cauchy - Riemann relations 3 Cauchy integral formula 9 Cauchy integrals 283-288 Cauchy's theorems 8 Causality 121 Chebyshev polynomials 335-338 Clenshaw's algorithm 343 Complementary error function 88 208 Continued fractions 350 Continuity of linear functionals 147 contour 2 Contour integration 6 Controllability 75 Convergence of generalized functions 150 Convergence 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 Pad 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 - K 239-242 Euler's constant 12 199 221 Exponential integral 12 Factorial function 19-22 Factorial function, asymptotic expansion 216 Factorial function, functional relations 20 Factorial function, Hankel's integral representation 22 Fast Fourier Transform 341 feedback loop 63 Fourier integrals, ascending expansions for 221 Fourier series 229 252 258 343 Fourier transform, application to partial differential equations 129-140 Fourier transform, definition 111 Fourier transform, in two or more variables 181 189 Fourier transform, inverse of 112 Fourier transform, of generalized functions 155 Fourier transform, of test functions 145 Fourier transform, properties of 116-118 Fourier transform, relation to Green's functions 254 Fourier transform, relation to Hankel transform 229 Fourier transform, relation to Laplace transform 111 Fourier transform, sine and cosine transforms 112 Fractional integration 239 Fraunhofer diffraction 186 Fresnel diffraction 186 Functional 144 Functional, analytic 153-156 Functional, continuous 147 Functional, linear 146 Functional, regular 147 Functional, singular 147 Generalized functions 143-157 167 276 Generalized functions, convergence of 150 Generalized functions, definition 146 Generalized functions, differentiation of 149 Generalized functions, Fourier transforms of 155 Generalized functions, on finite interval 148 Generalized functions, properties of 147-151 Generalized functions, regular 147 Generalized functions, sequences of 150 Generalized functions, singular 147 Green's functions 163-166 Green's functions, as generalized functions 167 Green's functions, for adjoint 165 Green's functions, for Helmholtz's equation 173 Green's functions, for Poisson's equation 169 Green's functions, integral transforms generated by 249 Green's functions, one-dimensional 163 Green's functions, symmetry of 171 Green's theorem 7 H 285 Hankel functions 173 228 272 317-320 Hankel transform 227 Hankel transform, application to boundary-value problems 232 Hankel transform, connection with Fourier transform 229 Hankel transform, definition 227 Hankel transform, inverse of 227 Hankel transform, properties of 230 231 Hankel transform, 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 Heaviside expansion theorem 48 Heaviside series expansion 53 Heaviside step function 28 Helmholtz's equation 173-176 266 Helmholtz's equation, elementary solution 173 Helmholtz's equation, Green's function for 176 Hermite equation 305 Hermite functions 307-310 Hermite functions, asymptotic forms 205-207 Hermite polynomials 305-307 Hopf 265 Hydrodynamic equations 132 Images 172 Impedance 93 Influence function 122 Integral equations 97-107 274 292 Integral equations, classification 97 Integral equations, dual 236-239 Integrals, Fourier 221 Integrals, involving a parameter 218 Integro-differential equations 274 Inverse Fourier transform 112 Inverse Fourier transform, sine and cosine transform 113 Inverse Laplace transform 39 Inverse Laplace transform, asymptotic forms of 50-54 Inverse Laplace transform, involving a branch point 49 Inverse Laplace transform, numerical evaluation of 327-355 Inverse Laplace transform, of meromorphic functions 47 Inverse Laplace transform, of rational functions 44 Inverse Laplace transform, Taylor series of 46 Jacobi polynomials 335 Kirchhoff 185 Kontorovich - Lebedev transform 256-262 Kontorovich - Lebedev transform, relation to Mellin transform 258 Kramers - Kr 121 Lagrangian interpolation 333 Laguerre polynomials 210 334 338 Laplace transform, application to ordinary differential equations 59 Laplace transform, application to partial differential equations 85-93 Laplace transform, application to simultaneous differential equations 67 Laplace transform, asymptotic properties 33 52 Laplace transform, definition 27 Laplace transform, differential equations with polynomial coefficients 65 Laplace transform, double 192-194 Laplace transform, inverse of 39 Laplace transform, inversion theorem 42 Laplace transform, properties of 28-32 Laplace transform, relation to Fourier transform 111 Laplace transform, Watson's lemma 35 50 Laplace's equation 129 190 233 Laplace's method 66 303-321 Laurent expansion 13 Lienard - Wiechert potential 189 Linear control theory 72-82 Linear control theory, controllability 75 Linear control theory, equivalent systems 77 Linear control theory, minimal realization 79 Linear control theory, observability 78 Linear control theory, realization 79 Linear functionals 146 Linear transport theory 291-297 Liouville's theorem 18 Lommel's integral 228 Loop integrals 15 49-53 M 331 MacDonald's function 321 MacRobert 227 Matrix exponential 73 Maxwell's equations 187 Mellin transform 195 Mellin transform, application to differential equations 205 Mellin transform, application to potential problems 202 Mellin transform, definition 195 Mellin transform, in asymptotics 197-200 215- 222 Mellin transform, in summation 211-216 Mellin transform, inverse of 196 Mellin transform, properties of 200-203 Mellin transform, relation to Fourier transform 195 Mellin transform, relation to Green's functions 255 Meromorphic functions 14 47 Meromorphic functions, inverse Laplace transform of 47 Method of images 172 Milne's equation 274 284 Minimal realization of transfer function 79 Mittag - LefHer theorem 281 Modified Bessel functions 320 Newton's law of cooling 89 Newton's second law 132 Normal system 71 Numerical inversion of Laplace transforms 327-355 Numerical inversion of Laplace transforms, collocation methods 333 Numerical inversion of Laplace transforms, Fourier series methods 343 Numerical inversion of Laplace transforms, Gaver - Stehfest method 329 Numerical inversion of Laplace transforms, Korrectur method 346 Numerical inversion of Laplace transforms, Lyness and Giunta's method 340 Numerical inversion of Laplace transforms, method of de Hoog, Knight, and Stokes 349 Numerical inversion of Laplace transforms, Talbot's method 352 Numerical inversion of Laplace transforms, Weeks's methods 339 Observability 78 Ordinary differential equations, Green's functions for 163-169 Ordinary differential equations, Laplace transform methods for 57-77 79-82 Ordinary differential equations, Laplace's method for 303-321 Ordinary differential equations, stability of solutions 60 Pad 349 Pair distribution function 104 Parseval relations 118 121 231 Partial differential equations, Fourier transform methods for 129-140 Partial differential equations, 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- 202 233-237 Power series 10 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 - Hilbert problem 289-291 Riemann zeta function 23-26 Riemann zeta function, asymptotic forms 26 Riemann zeta function, functional relation 25 Riemann zeta function, in summation 212 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 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 Watson's lemma, for loop integrals 50-53 Wave equation 85 90 173 185 259 265 297 Wave propagation 90
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