Asymptotic and special functions

Title page......Page 1
Preface to AK Peters Edition......Page 11
Preface......Page 13
1 Origin of Asymptotic Expansions......Page 14
2 The Symbols ~, o, and O......Page 17
3 The Symbols ~, o, and O (continued)......Page 19
4 Integration and Differentiation of Asymptotic and Order Relations......Page 21
5 Asymptotic Solution of Transcendental Equations: Real Variables......Page 24
6 Asymptotic Solution of Transcendental Equations: Complex Variables......Page 27
7 Definition and Fundamental Properties of Asymptotic Expansions......Page 29
8 Operations with Asymptotic Expansions......Page 32
9 Functions Having Prescribed Asymptotic Expansions......Page 35
10 Generalizations of Poincaré's Definition......Page 37
11 Error Analysis; Variational Operator......Page 40
Historical Notes and Additional References......Page 42
1 The Gamma Function......Page 44
2 The Psi Function......Page 52
3 Exponential, Logarithmic, Sine, and Cosine Integrals......Page 53
4 Error Functions, Dawson's Integral, and Fresnel Integrals......Page 56
5 Incomplete Gamma Functions......Page 58
6 Orthogonal Polynomials......Page 59
7 The Classical Orthogonal Polynomials......Page 61
8 The Airy Integral......Page 66
9 The Bessel Function J_ν(z)......Page 68
10 The Modified Bessel Function I_ν(z)......Page 73
11 The Zeta Function......Page 74
Historical Notes and Additional References......Page 77
1 Integration by Parts......Page 79
2 Laplace Integrals......Page 80
3 Watson's Lemma......Page 84
4 The Riemann-Lebesgue Lemma......Page 86
5 Fourier Integrals......Page 88
6 Examples; Cases of Failure......Page 89
7 Laplace's Method......Page 93
8 Asymptotic Expansions by Laplace's Method; Gamma Function of Large Argument......Page 98
9 Error Bounds for Watson's Lemma and Laplace's Method......Page 102
10 Examples......Page 105
11 The Method of Stationary Phase......Page 109
12 Preliminary Lemmas......Page 111
13 Asymptotic Nature of the Stationary Phase Approximation......Page 113
Historical Notes and Additional References......Page 117
1 Laplace Integrals with a Complex Parameter......Page 119
2 Incomplete Gamma Functions of Complex Argument......Page 122
3 Watson's Lemma......Page 125
4 Airy Integral of Complex Argument; Compound Asymptotic Expansions......Page 129
5 Ratio of Two Gamma Functions; Watson's Lemma for Loop Integrals......Page 131
6 Laplace's Method for Contour Integrals......Page 134
7 Saddle Points......Page 138
8 Examples......Page 140
9 Bessel Functions of Large Argument and Order......Page 143
10 Error Bounds for Laplace's Method; the Method of Steepest Descents......Page 148
Historical Notes and Additional References......Page 150
1 Existence Theorems for Linear Differential Equations: Real Variables......Page 152
2 Equations Containing a Real or Complex Parameter......Page 156
3 Existence Theorems for Linear Differential Equations: Complex Variables......Page 158
4 Classification of Singularities; Nature of the Solutions in the Neighborhood of a Regular Singularity......Page 161
5 Second Solution When the Exponents Differ by an Integer or Zero......Page 163
6 Large Values of the Independent Variable......Page 166
7 Numerically Satisfactory Solutions......Page 167
8 The Hypergeometric Equation......Page 169
9 The Hypergeometric Function......Page 172
10 Other Solutions of the Hypergeometric Equation......Page 176
11 Generalized Hypergeometric Functions......Page 181
12 The Associated Legendre Equation......Page 182
13 Legendre Functions of General Degree and Order......Page 187
14 Legendre Functions of Integer Degree and Order......Page 193
15 Ferrers Functions......Page 198
Historical Notes and Additional References......Page 202
1 The Liouville Transformation......Page 203
2 Error Bounds: Real Variables......Page 206
3 Asymptotic Properties with Respect to the Independent Variable......Page 210
4 Convergence of V(F) at a Singularity......Page 213
5 Asymptotic Properties with Respect to Parameters......Page 216
6 Example: Parabolic Cylinder Functions of Large Order......Page 219
7 A Special Extension......Page 221
8 Zeros......Page 224
9 Eigenvalue Problems......Page 227
10 Theorems on Singular Integral Equations......Page 230
11 Error Bounds: Complex Variables......Page 233
12 Asymptotic Properties for Complex Variables......Page 236
13 Choice of Progressive Paths......Page 237
Historical Notes and Additional References......Page 241
1 Formal Series Solutions......Page 242
2 Asymptotic Nature of the Formal Series......Page 245
3 Equations Containing a Parameter......Page 249
4 Hankel Functions; Stokes' Phenomenon......Page 250
5 The Function Y_ν{z)......Page 254
6 Zeros of J_ν(z)......Page 257
7 Zeros of Y_ν{z) and Other Cylinder Functions......Page 261
8 Modified Bessel Functions......Page 263
9 Confluent Hypergeometric Equation......Page 267
10 Asymptotic Solutions of the Confluent Hypergeometric Equation......Page 269
11 Whittaker Functions......Page 273
12 Error Bounds for the Asymptotic Solutions in the General Case......Page 275
13 Error Bounds for Hankel's Expansions......Page 279
14 Inhomogeneous Equations......Page 283
15 Struve's Equation......Page 287
Historical Notes and Additional References......Page 290
1 The Euler-Maclaurin Formula and Bernoulli's Polynomials......Page 292
2 Applications......Page 297
3 Contour Integral for the Remainder Term......Page 302
4 Stirling's Series for ln Γ(z)......Page 306
5 Summation by Parts......Page 308
6 Barnes' Integral for the Hypergeometric Function......Page 312
7 Further Examples......Page 315
8 Asymptotic Expansions of Entire Functions......Page 320
9 Coefficients in a Power-Series Expansion; Method of Darboux......Page 322
10 Examples......Page 324
11 Inverse Laplace Transforms; Haar's Method......Page 328
Historical Notes and Additional References......Page 334
1 Logarithmic Singularities......Page 335
2 Generalizations of Laplace's Method......Page 338
3 Example from Combinatoric Theory......Page 342
4 Generalizations of Laplace's Method (continued)......Page 344
5 Examples......Page 347
6 More General Kernels......Page 349
7 Nicholson's Integral for J²_ν(z) + Y²_ν(z)......Page 353
8 Oscillatory Kernels......Page 355
9 Bleistein's Method......Page 357
10 Example......Page 359
11 The Method of Chester, Friedman, and Ursell......Page 364
12 Anger Functions of Large Order......Page 365
13 Extension of the Region of Validity......Page 371
Historical Notes and Additional References......Page 374
1 Classification and Preliminary Transformations......Page 375
2 Case I: Formal Series Solutions......Page 377
3 Error Bounds for the Formal Solutions......Page 379
4 Behavior of the Coefficients at a Singularity......Page 381
5 Behavior of the Coefficients at a Singularity (continued)......Page 382
6 Asymptotic Properties with Respect to the Parameter......Page 384
7 Modified Bessel Functions of Large Order......Page 387
8 Extensions of the Regions of Validity for the Expansions of the Modified Bessel Functions......Page 391
9 More General Forms of Differential Equation......Page 395
10 Inhomogeneous Equations......Page 399
11 Example: An Inhomogeneous Form of the Modified Bessel Equation......Page 401
Historical Notes and Additional References......Page 404
1 Airy Functions of Real Argument......Page 405
2 Auxiliary Functions for Real Variables......Page 407
3 The First Approximation......Page 410
4 Asymptotic Properties of the Approximation; Whittaker Functions with m Large......Page 414
5 Real Zeros of the Airy Functions......Page 416
6 Zeros of the First Approximation......Page 418
7 Higher Approximations......Page 421
8 Airy Functions of Complex Argument......Page 426
9 Asymptotic Approximations for Complex Variables......Page 429
10 Bessel Functions of Large Order......Page 432
11 More General Form of Differential Equation......Page 439
12 Inhomogeneous Equations......Page 442
Historical Notes and Additional References......Page 446
1 Bessel Functions and Modified Bessel Functions of Real Order and Argument......Page 448
2 Case III: Formal Series Solutions......Page 451
3 Error Bounds: Positive ζ......Page 453
4 Error Bounds: Negative ζ......Page 456
5 Asymptotic Properties of the Expansions......Page 460
6 Determination of Phase Shift......Page 462
7 Zeros......Page 464
8 Auxiliary Functions for Complex Arguments......Page 466
9 Error Bounds: Complex u and ζ......Page 470
10 Asymptotic Properties for Complex Variables......Page 473
11 Behavior of the Coefficients at Infinity......Page 475
12 Legendre Functions of Large Degree: Real Arguments......Page 476
13 Legendre Functions of Large Degree: Complex Arguments......Page 483
14 Other Types of Transition Points......Page 487
Historical Notes and Additional References......Page 491
2 Connection Formulas at a Singularity......Page 493
3 Differential Equations with a Parameter......Page 495
4 Connection Formula for Case III......Page 496
5 Application to Simple Poles......Page 500
6 Example: The Associated Legendre Equation......Page 503
7 The Gans-Jeffreys Formulas: Real-Variable Method......Page 504
8 Two Turning Points......Page 507
9 Bound States......Page 510
10 Wave Penetration through a Barrier I......Page 514
11 Fundamental Connection Formula for a Simple Turning Point in the Complex Plane......Page 516
12 Example: Airy's Equation......Page 520
13 Choice of Progressive Paths......Page 521
14 The Gans-Jeffreys Formulas: Complex-Variable Method......Page 523
15 Wave Penetration through a Barrier II......Page 526
Historical Notes and Additional References......Page 529
1 Numerical Use of Asymptotic Approximations......Page 532
2 Converging Factors......Page 535
3 Exponential Integral......Page 536
4 Exponential Integral (continued)......Page 540
5 Confluent Hypergeometric Function......Page 544
6 Euler's Transformation......Page 549
7 Application to Asymptotic Expansions......Page 553
Historical Notes and Additional References......Page 556
Answers to Exercises......Page 558
References......Page 561
Index of Symbols......Page 574
General Index......Page 576