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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Bender C., Orszag S. — Advanced Mathematical Methods for Scientists and Engineers |
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| Ïðåäìåòíûé óêàçàòåëü |
Parametric amplifier 568p
Patching 335—336 497—504 500—505f 527
Period T of Duffing’s equation 548 567p
Period T of Rayleigh oscillator 477—479
Periodic behaviors for difference equations 238 246p
Periodic behaviors, asymptotic relations for 107—112 150—152 161—164
Periodic behaviors, Floquet theory 560—561 568p
Periodic behaviors, limit cycles see “Limit cycles”
Periodic behaviors, oscillators see “Oscillators”
Periodic behaviors, oscillatory integrals see “Stationary phase method
Periodic behaviors, resonant and secular behavior 544—545
Periodic behaviors, slowly varying frequency 556 558—559
Perturbation problems, regular, initial-value problems 321—322 323f 327 329f
Perturbation problems, regular, roots of polynomials 319—320 328—330 330t 331f
Perturbation problems, regular, Schrodinger eigenvalue problem 330—335 354—355
Perturbation problems, singular, boundary-layer see “Boundary-layer problems”
Perturbation problems, singular, boundary-layer problems solved by asymptotic matching 337—341
Perturbation problems, singular, evaluation of integrals using asymptotic matching 341—349 365—366p 515—519
Perturbation problems, singular, initial-value problems solved by asymptotic matching 336—337
Perturbation problems, singular, multiple-scale see “Multiple-scale perturbation problems”
Perturbation problems, singular, roots of polynomials 325—326 361p
Perturbation problems, singular, Schrodinger eigenvalue problem 334—335 353 359—360
Perturbation problems, singular, WKB see “WKB problems”
Perturbation series 319—367
Perturbation series for eigenvalues and eigenfunctions 333—335 363p
Perturbation series in boundary-layer problems 428—429 431—435 440—455 477—479
Perturbation series in multiple-scale problems 545—551 561—566
Perturbation series in WKB problems 486—487 493—494 496 534—539
Perturbation series, derivation from asymptotic matching 337 342—349
Perturbation series, introductory discussion 317—318
Perturbation series, summation of 318 368—415 547—549
Perturbation series, Taylor series vs. 66 321—323 323f
Perturbation series, zeroth-order (leading) term in 320—321 324
Perturbation theory for linear eigenvalue problems 330—335
Perturbation theory, asymptotic matching 335—349
Perturbation theory, boundary-layer theory 326 419—484
Perturbation theory, dissipative and dispersive phenomena in 484—486
Perturbation theory, general procedures of 319—323
Perturbation theory, local and global breakdown in 484—485
Perturbation theory, mathematical structure of perturbative eigenvalue problems 350—361
Perturbation theory, multiple-scale 327—328 329f 540—568
Perturbation theory, philosophy of 317—318 368
Perturbation theory, regular, definition of 324
Perturbation theory, singular, definition of 324
Perturbation theory, unperturbed problem 321 324 333
Perturbation theory, WKB theory 326—327 328f 484—543
Phase space 171—197
Phase space, asymptotic matching in 337
Phase space, definition of 172
Phase space, higher-dimensional 185—197
Phase space, phase line 174—175
Phase space, phase plane 175—185 468—479 546—547
Phase space, phase sphere 202—203p 203f
Phase space, volume in 190—191 204p
Physical optics approximation as leading asymptotic behavior of WKB series 494—533
Physical optics approximation for Green’s functions 497—504 500—505f
Physical optics approximation for higher-order equations 496—497
Physical optics approximation in complex plane 541p 542p
Physical optics approximation, accuracy of 494—497
Physical optics approximation, definition of 494—495
Physical optics approximation, derived using multiple-scale theory 556 558—559
Physical optics approximation, failure of 495—496
Physical optics approximation, failure of, at turning points 497 568p
Physical optics approximation, integrals of 490—491 500—504 513—519 540—541p
Physical optics approximation, normalization of 490 513—519
Physical optics approximation, validity of 494—497 506—508 521
Poincare plot 188 189f 190f
Polynomials, Bernoulli 305 315p
Polynomials, finding roots of 234—235 24lp 319—320 325—326 328—330 330t 331f 332f 352 354 361p 363p 366p
Polynomials, Hermite 55 99 133 244p
Polynomials, Laguerre 244p
Polynomials, Legendre 55p 229 231 232t 243p 309p
Potential 331 520—521 521f 526—527
Potential barriers,  -function 526—527
Potential barriers, scattering off peak of 531—533
Potential barriers, tunneling through 524—531
Predatory-prey systems 179—183 180—182f
probability density 543p
Psi function  see “Digamma function”
Quantization condition, WKB see “Eigenvalue condition”
Quantum anharmonic oscillator 334—335 337—338 353—354 353t 359—360 366—367p
Quantum harmonic oscillator 28 133 332—333 522—533 538
Quasiclassical approximation see “WKB theory”
Radioactive decay 543p
Ramp function 17
Random behaviors , Poincare plot 188 189f 190f
Random behaviors in phase space 188—189 190f 192—195 195—197f
Random behaviors of difference equations 237—239 239t
Rapid variation on global scale 326—327 328f 493
Rapid variation on local scale (at boundary layers) 326 327f 493
Rapid variation, definition of 484—485
Rayleigh oscillator, boundary-layer analysis of 468—479 476—478f 483p
Rayleigh oscillator, multiple-scale analysis of 554—556 556f 557f
Rayleigh oscillator, period of 477—479
Rayleigh —Ritz variational procedure 352
Recessive see “Subdominance”
Recursion (recurrence) relations see “Difference equations”
Reduction of order for difference equations, homogeneous linear 43—45 56p
Reduction of order for difference equations, inhomogeneous linear 51—52 55p 56p
Reduction of order for difference equations, using generating functions 46
Reduction of order for differential equations, homogeneous linear 13 32p
Reduction of order for differential equations, inhomogeneous linear 19 32p 333
Reduction of order for differential equations, nonlinear 24—27 34p 35p 156
Reflection coefficient R 527 531 533
Regular perturbation theory see “Perturbation theory”
Regular singular points, of difference equations, definition of 207
Regular singular points, of difference equations, Frobenius series about 212—214
Regular singular points, of difference equations, leading asymptotic behavior near 207—210
Regular singular points, of differential equations, definition of 62 64
Regular singular points, of differential equations, Frobenius series near 68—76 452
Regular singular points, of differential equations, local behavior near 68—76 452—453
Regular singular points, of differential equations, Taylor series near 63
Regular singular points, of differential equations, Taylor series near, failure of 68—69
Resonant behavior 544—545 (see also “Secular behavior”)
Resonant scattering 542—543p
Riccati differential equation, conversion to second-order linear 21 151 156 483p
Riccati differential equation, definition of 20
Riccati differential equation, reduction to Bernoulli equation 21
Riccati differential equation, soluble and insoluble 4 22 27
Riccati differential equation, spontaneous singular points of 149—152 150f 152f 153t 197—198p
Richardson extrapolation 375—376 377t 378t 379 411—412p
Riemann integrals, approximating sums by 303—305
Riemann zeta function see “Zeta function”
Riemann — Lebesgue lemma 277—278 285 311p
Roots of polynomials, finding, Bernoulli’s method for 241p
Roots of polynomials, finding, Hermite polynomials 244p
Roots of polynomials, finding, Laguerre polynomials 244p
Roots of polynomials, finding, Legendre polynomials 231 232t
Roots of polynomials, finding, Newton’s method for 234—235
Roots of polynomials, finding, perturbation theory for 319—320 325—326 328—330 330t 331f 332f 361p 363p
Roots of polynomials, finding, Sylvester’s eliminant for 352 354 366p
Saddle points, in autonomous systems, one-dimensional 174—175 174f
Saddle points, in autonomous systems, two-dimensional 177 179—182 180—182f
Saddle points, in method of steepest descents, approximation to integrals with 290—300
Saddle points, in method of steepest descents, definition of 288
Saddle points, in method of steepest descents, movable 295
Saddle points, in method of steepest descents, second-, third-, and fourth-order 289—290 290—292f
Scale-invariant equations 25—26 156
Scattering off peak of potential barrier 531—533 542p
Scattering problems 524—533 542—543p
Schrodinger equation, eigenvalue problem, analytic structure of eigenvalues as functions of perturbation parameter 351—361
Schrodinger equation, eigenvalue problem, eigenvalues for 27—29 330—335
Schrodinger equation, eigenvalue problem, perturbative solution of 330—335
Schrodinger equation, eigenvalue problem, WKB solution of 519—523 523t 524—52 542p 543p
Schrodinger equation, leading asymptotic behavior of nth-order 88 102—103 103f 496—497
Schrodinger equation, reduction of general second-order linear differential equation to 32p
Schrodinger equation, time-dependent 524—525 543p
Schrodinger equation, turning points of 497 521 521f
Schrodinger equation, WKB approximation for second-order 486—488 489 494—496 504—539
Sector of validity for asymptotic relations 113—118 114f 115f 130—136
| Secular behavior, definition of 545
Secular behavior, elimination of 550 553 555 558—559 564 566 566—567p
Secular behavior, summation of leading secularities 547—549
Semiclassical approximation see “WKB theory”
separable differential equations 3 23 24 30p
Shanks transformation, definition of 370
Shanks transformation, Euler — Maclaurin sum formula vs. 379
Shanks transformation, examples of 370—375 370t 371f 373t 374f
Shanks transformation, failure of 375 381
Shanks transformation, generalized (higher-order) 389—392 412p
Shanks transformation, iterated 370—374
Shanks transformation, roundoff error 372 410—411p
Simpson’s rule 411—412p
Singular perturbation theory see “Perturbation theory”
Singular points of difference equations, definition and classification of 206—207
Singular points of difference equations, local analysis near 207—210 212—218 223—233
Singular points of differential equations, at  64
Singular points of differential equations, classification of 62—66
Singular points of differential equations, fixed 146 158
Singular points of differential equations, irregular 63—65 76—103
Singular points of differential equations, local behavior near 68—103
Singular points of differential equations, regular 62—63 68—76 452—453
Singular points of differential equations, removing 66
Singular points of differential equations, spontaneous (movable) 7 146—153 158 197—198p 554f
Slow variation 484—485
Snowplow problem 33p
Sommerfeld contour 293 293f 298
Spiral points 176—179 181—184 181f 182f 184f 192—194f 556f 557f
Spontaneous (movable) singular points for first Painleve transcendent 158
Spontaneous (movable) singular points for Riccati equations 149—152 150f 152f 153t 197—198p
Spontaneous (movable) singular points in complex plane 149
Spontaneous (movable) singular points, definition and discussion of 7 146—153
Spontaneous (movable) singular points, infinite number of 150—153
Spontaneous (movable) singular points, which are not poles 164—166
Stability of critical points, higher-dimensional 186 189 191
Stability of critical points, one—and two-dimensional 174—177
Stability of first Painleve transcendent 161 163—164
Stability, Floquet theory 560—561
Stabilityof solutions to Mathieu equation 561—566 568p
Stationary phase, method of 276—280 286 311—312p 347
Stationary points of order p 278—280
Stationary points, infinite-order 285—287
Steepest descents, method of 280—302
Steepest descents, method of, expanding Fourier integrals using 281—287 291 293—294
Steepest descents, method of, for complex parameter 300—302 313—315f7
Steepest descents, method of, for integrals with saddle points 290—300
Steepest descents, method of, steepest-descent and constant-phase paths in 282—287
Steepest descents, method of, use of, where Laplace’s method fails 296—300
Steepest-ascent paths see “Steepest-descent paths”
Steepest-descent paths, definition and formal discussion of 282—283 282f 287—289
Steepest-descent paths, introduction to use of 282—287
Steepest-descent paths, rotation of in complex plane 301—302 301f 302f method
Step function 16 18
Stieltjes function (Stieltjes integral), asymptotic expansion of 260
Stieltjes function (Stieltjes integral), continued-fraction representation of 396 406—407
Stieltjes function (Stieltjes integral), definition and properties of 120—122 406
Stieltjes function (Stieltjes integral), moment integrals 121—122 405
Stieltjes function (Stieltjes integral), sum of Stieltjes series 120—122
Stieltjes function (Stieltjes integral), weight (density) function 121—122 405—406
Stieltjes series as asymptotic series 78 120—123 143p 260
Stieltjes series, Borel sum of 406
Stieltjes series, Carleman condition 410 415p 416p
Stieltjes series, derivation of 77—78 260
Stieltjes series, differential equation for 77—78
Stieltjes series, error estimates for optimally truncated 123 124f
Stieltjes series, moment problem 410 414—415p
Stieltjes series, Pade approximants for 387—388 390t 404—410 415—416p
Stirling series, Bernoulli numbers and 242p
Stirling series, derivation of using difference equations 222—223 225—227
Stirling series, derivation of using Laplace’s method 275—276 310p
Stirling series, derivation of using method of steepest descents 294—296
Stirling series, description of 218—220 575
Stirling series, error estimates for 242p
Stirling series, optimal asymptotic approximation for 221t 222 223f 224t 242p
Stirling series, Pade approximants for 388—389 391
Stirling series, Richardson extrapolation of 376 377t
Stirling series, Taylor series vs. 220 221t 222
Stirling series, use of, to find minimum of  222
Stokes lines, definition of 116 (see also “Stokes phenomenon”)
Stokes phenomenon for Airy functions 116—118
Stokes phenomenon for parabolic cylinder functions 117
Stokes phenomenon in WKB theory 541p
Stokes phenomenon, definition of 115—116
Stokes phenomenon, functional relations used to determine 130—133 143p
Stokes phenomenon, steepest-descents method used to determine 300—302 313—315p
Sturm — Liouville problem 29 35p 351 490—491 491f 492f 492t 540p
Subdominance in boundary-layer problems 429
Subdominance in Laplace’s method 262—263 273—274
Subdominance in WKB problems 531 (see also “Stokes phenomenon”)
Subdominance, explanation of 115—118
Substitutions for solving difference equations 53
Substitutions for solving differential equations 23—24 151 156—157
Summation methods 368—415
Summation methods for convergent series 368—379 389—392'
Summation methods for divergent series 379—410
Summation methods, asymptotic summation of series 376 379 380t
Summation methods, Borel 381—382
Summation methods, Euler 381
Summation methods, generalized (two-point) Pade 393—394
Summation methods, generalized Shanks transformation 389—392
Summation methods, multiple-scale methods as 549
Summation methods, optimal asymptotic approximation 94—100 112 112f 122—124 124f 221—224 376 379 388 392
Summation methods, Pade 383—410
Summation methods, philosophy of 120 368 379—381
Summation methods, Richardson extrapolation 375—376 377t 378t
Summation methods, Shanks transformation 369—375
Summation methods, summation of leading secularities 547—549 (see also “Sums asymptotic
Summing factor 39
Sums, asymptotic evaluation of 302—306
Sums, asymptotic evaluation of approximation by Riemann integrals 303—304
Sums, asymptotic evaluation of Euler — Maclaurin sum formula for 305—306 306t 315 379 411p
Sums, asymptotic evaluation of Laplace’s method applied to 304—305
Sums, asymptotic evaluation of truncation of 303
Sylvester’s eliminant 352 354 366p
Systems of first-order differential equations, autonomous see “Autonomous systems”
Systems of first-order differential equations, equivalence of, to general nth-order differential equation 5
Systems of first-order differential equations, removing a singularity by conversion to 66
Tauberian theorems 127—128 145p
Taylor series at ordinary points 62 65—70 104
Taylor series for  220 221t 222t 254f 255 385 387t
Taylor series for Airy functions 67—68 569
Taylor series for Bessel functions 143p
Taylor series for difference equations 210—212
Taylor series for digamma function 575
Taylor series for nonlinear differential equations 147—149
Taylor series for parabolic cylinder functions 574
Taylor series, asymptotic series vs. 90—91 92f 118—119 220 221t 222 254f 255
Taylor series, convergence of 90—91 92f
Taylor series, derivation of 252—253
Taylor series, leading asymptotic behaviors of 90—92 140p 304—305
Taylor series, near irregular singular points 65
Taylor series, near regular singular points 63 68—69
Taylor series, Pade approximants vs. 384—387 385—388t 400
Taylor series, perturbation series vs. 66 321—323 323f
Taylor series, Shanks transformation of 369—372 370t 371f
Taylor series, slowly converging 368—372
Taylor series, transients in 369—371
Thomas — Fermi equation 25—26 167—170 168—169t 170f 199—200p
Time scales, long and short 549 556 558—559
Time scales, multiple 551 564 567p
Toda lattice 187 188f 190 203p
Trajectories in phase space, almost periodic orbits 186—187 188f 189f
Trajectories in phase space, cycles 173f 175
Trajectories in phase space, definition of 172—173
Trajectories in phase space, global analysis of, in higher dimensions 185—197
Trajectories in phase space, global analysis of, in one dimension 174—175 176f
Trajectories in phase space, global analysis of, in two dimensions 175 178—183 468—470
Trajectories in phase space, global analysis of, using boundary-layer theory 468—479 476—478f 483p
Trajectories in phase space, limit cycles 183 184f 468—479 476—478f 483p 544—556 556f 557f
Trajectories in phase space, local analysis of, near critical points in higher dimensions 185—197
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