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| Ïîèñê ïî óêàçàòåëÿì |
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| Weinberger H.F. — First course in partial defferential equations with complex variables and transform methods |
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| Ïðåäìåòíûé óêàçàòåëü |
Absolute value 204
Absolutely integrable function 305 306
Analytic at infinity 278
Analytic continuation 216
Analytic extension 215
Analytic function 103 213
Approximation, in the mean 70 72
Approximation, in the mean, pointwise 70
Argand diagram 201
Argument 204
Associated Legendre functions 192 193
Asymptotic behavior of the Laplace transform 349 350
Bessel function 149 179—181 340—344
Bessel function with imaginary argument 149 340—344
Bessel’s equation 176 179
Bessel’s inequality 73
Bilinear transformation 249
Boundary mesh point 380
Boundary value problem, two-point 120—126 168
Branch cut 227 293—296
Calculus of residues 282—292 303 304
Cauchy criterion 209 307
Cauchy integral representation 264 266
Cauchy principal value 284 287 290 291 313 332
Cauchy-Riemann equations 211 213 223 237
Cauchy’s Theorem 220 262
Chain rule for analytic functions 234
Change of scale 88—92 95
Characteristic (natural) frequency 154 186 187
Characteristic cone 153
Characteristic coordinates 46
Characteristic surface 154
Characteristic triangle 26
Characteristic values (eigenvalues) 65 124 160—168 171—178 180 186 187 391
Characteristics 18—21 38 42 43 45 59 115
Circle of convergence 210
Classification of partial differential operators 41—46 51 52 59
Closed curve 48 49
Complete set of functions 74 75 84 85 166 176—178
Completeness of eigenfunctions 166 176—178
Complex conjugate 203
complex derivative 214
complex numbers 201—206
Complex path integral 219
Conformal mapping 238—245 257—261
Conjugate harmonic function 212
Connected set 218
Conservation of energy 36
Continuity with respect to data 6 15 34 56 61 107 318 382
Contour integrals 220 261 262 263 271—277 282—296
convergence, in the mean 71 84 85 143 166 307 308 309
Convergence, in the mean, of Fourier series 79 80 81—88
Convergence, in the mean, pointwise 70
Convergence, in the mean, uniform 33 70 72 81 83 84 145 167 177 178 307
Convolution product 326 347
Convolution theorem 327 331 340 347
Cosine series 67 87 88
Cosine transform 321
Damped wave equation 112—115
Derivative of an analytic function 214
Derivatives of Bessel functions 179 180
Descent, method of 336
Difference quotient 375
Diffraction 369
Diffusion 59
Dimensional analysis 91
Dini’s test 79 80
Dirichlet’s principle 393 396
Discontinuities, propagation of 21 40 42 43 46 115 154
Divergence theorem 52 58
Domain 49 218
Domain of dependence 18 23 26 37 39 42 46 115 120 377 387 388
Domain of influence 20 23 26 40 42 46
Double Fourier series 141—145
Duhamel’s principle 371
d’Alembert solution 9 13 26 114
Eigenfunction 65 160—168 171—178 186
Eigenvalue 65 124 160—168 171—178 180 186 187 391
Elastic bar 6 91
Electrostatic potential 49 50
Elliptic coordinates 193
Elliptic differential operator 43 44 51 52
Energy 36 53 110 152—154
Entire function 217 266
Error bounds 86 97 98 102 145 266 386 389
Essential singularity 271
Existence 6
Exponential function 215 216 257
Extended plane 246
Finite difference approximation 375—383
Finite Fourier transform 129 130 186 197
Finite sine transform 127
Fixed boundary condition 23
Focusing 154
Fourier coefficients 72
Fourier cosine series 67 87 88
Fourier inversion theorems 314 315 317 321 332 338 359
Fourier series 65 66 67 77—88 141—145 298
Fourier sine series 65 66 67 87 88
Fourier transform 301 310—313 329—332 337—345
Free boundary condition 23
Frobenius method 179
Gauss’s theorem (divergence theorem) 52 58
Generalized circle 247
Green’s function 122—125 135—140 158 161 167 177 178 197 240 241
Green’s function, one-sided 119
Green’s Theorem 53 59
Harmonic conjugate 212
Harmonic function 52
Harmonic polynomials 104
Heat equation 58 60 66 92—95 108—110 126 318 322 327 329—332 355—361
Heat flow 50 58
Heat kernel 328
Heaviside function 347 367
Holder continuity 79 84
Holomorphic function 213
Homogeneous equation 30 31
Huyghen’s principle 336
Hyperbolic differential operator 42 44
Hyperbolic functions 227 258 259 260
Imaginary part 201
Implicit function theorem 230
Improperly posed problem 51 61
Infinite series evaluated by residues 289—292
Influence function 119 352
Initial value problems for ordinary differential equations 117—120
Integral equation 385
Integral of an analytic function 222
Interior mesh point 380
Inverse Fourier transform 314 315 317 319 321 332 338
Inverse function 229 230 239
Inverse Laplace transform 348
Inverse points 247 248
Inverse transformation 229 230 239
inversion 246 247
Isolated singularity 269
Iteration 383 384—391
Jacobian determinant 25 230
Jordan’s lemma 303
Kirchhoff’s formula 335
| Laplace operator 49
Laplace operator in cylindrical coordinates 149
Laplace operator in polar coordinates 100
Laplace operator in spherical coordinates 188
Laplace transform 346 349 350
Laplace’s equation 43 49 63 95—107 110—111 149—151 194—196 236—245 253 380—383
Laplace’s solution of the heat equation 331
Laurent series 278—281
Least squares approximation 70 72
Legendre functions, associated 192 193
Legendre polynomials 191 192 193
Limit of a complex sequence 208
Linear fractional transformation 249
Linear operator 29
Linear partial differential equation 30
Linear problem 30
Linear transformation 246
Liouville’s theorem 266 267
Logarithm 224 230
L’Hopital’s Rule 271
Mapping 238
Mapping, conformal 238—245 257—261
mapping, one-to-one 238
Mathematical model 5
Maximum principle 55—57 59 61 94 97 101 108 111 265 320 381
Mean convergence 71 84 85 143 166 307 308 309
Mean value theorem 103 196
Mean-square deviation 71
Mellin inversion theorem 348
Membrane 48 54 182—184
Mesh point 376
Method of descent 336
Method of successive approximations 384—391
Minimum principle for eigenvalues 164—165 168
Moebius transformation 249
Monotonicity of eigenvalues 173 174 175
Morera’s theorem 223
Multiple Fourier series 141—145
Multiple Fourier transform 329—332
Multiple-valued analytic function 223 224 227 293—296
Multiply connected domain 220
Natural (characteristic) frequencies 154 186 187
nearest neighbors 380
Nonhomogeneous wave equation 25
Normal modes 154 186 187
Null function 308 309 310
One-sided Green’s function 119
One-to-one mapping 238
Open set 49 218
Operational formulas (rules) 324—325 331 340 346 351
Operator 29
Order of a pole 270
Order of a zero 270
Ordinary differential equations 117—126 167—168 351—354
Orthogonal functions 71 143 161
Oscillation theorem 174 175
Parabolic differential operator 42 44 59
Parseval’s equation 74 75 85 142 166 177 312 313 322 332
Partial differential operator 29
path integral 219
Phragmen — Lindelof theorem 107 111 112
Point at infinity 246
Pointwise convergence 70
Poisson’s equation 48 49 50 129 155—158 196—199
Poisson’s integral formula 102 103 139 196 199 252
Poisson’s solution of the wave equation 335
Polar representation of a complex number 204 216
Pole 270
Power function 227 260
Power series 179 190 209—219
Principal value 284 287 290 291 313 332
Principle of superposition 33
Propagation of discontinuities 21 40 42 43 46 115 154
Properly posed problem 6 51 61 125 377 382
Radius of convergence 210 213 217
Ratio Test 213
Rayleigh quotient 163
Rayleigh — Ritz method 393—395
Real part 201
Reciprocity law 122
Recursion formulas for Bessel functions 179 180
Removable singularity 269
Residue 273 274
Residue theorem 275 283 284 285 290 303 304
Resonance 187
Riemann — Leuesgue lemma 76 316
Riesz — Fischer theorem 308 309
Rodrigues formula 191
Rouche’s Theorem 278
Scaling 88—92 95
Schwarz’s inequality 82 83 84 306
Self-adjoint form of an ordinary differential equation 117
Separable partial differential operator 63 66—68
Separation of variables 63—69 92—103 379
Separation theorem 173
Separation theorem for eigenvalues 175
Series evaluated by residues 289—292
sgn 433
Shift formula (shift rule) 324 340
Simple closed contour 220
Simply connected domain 49 220
Sine series 65—67 87 88
Sine transform 321
Singular differential equation 176 177
Singularity 217 269
Singularity, essential 271
Singularity, isolated 269
Singularity, removable 269
Sound waves 7 336 362
Specific heat 58
Spherical harmonics 192 193 195
Square integrable function 306 310
Stability condition 378 379
Stokes rule 370
Stokes’ Theorem 53
String 1 5
Strong maximum principle 265
Successive approximations 384—391
Superposition 33
Taylor series 214 215 231—235
Tchebysheff polynomials 73
Telegrapher’s equation 115
Temperature 50 58
Term-by-term integration of Fourier series 75
thermal conductivity 58
Thomson’s principle 397
Three-dimensional wave equation 152—154 333—336
Triangle inequality 205 307
Trigonometric Fourier series 77—88
Trigonometric functions of a complex variable 227 260
Two-dimensional wave equation 362 —371 378—379
Two-point boundary value problem 120—126 168
Unbounded domains 253—256
Uniform convergence 33 70 72 81 83 84 145 167 177 178 307
Uniqueness 6 34 39 53 56 59—61 64 107 110 152—155 318 382
Vibrating membrane 48 54 182—184
Vibrating string 1 5
Wave equation, damped 112—115
Wave equation, damped, one-dimensional 9—27 66
Wave equation, damped, three dimensional 152—154 333—336
Wave equation, damped, two-dimensional 362—371 378—379
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