Asymtotics and Special Functions

Автор(ы):Olver F. V. J.
06.10.2007
Год изд.:1974
Описание: Classical analysis is the backbone of many branches of applied mathematics. The purpose of this book is to provide a comprehensive introduction to the two topics in classical analysis mentioned in the title. It is addressed to graduate mathematicians, physicists, and engineers, and is intended both as a basis for instructional courses and as a reference tool in research work. It is based, in part, on courses taught at the University of Maryland.
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Asymtotics and Special Functions — обложка книги.
1 Introduction to Asymptotic Analysis
  1 Origin of Asymptotic Expansions [1]
  2 The Symbols ~, o, and О [4]
  3 The Symbols ~, o, and О (continued) [6]
  4 Integration and Differentiation of Asymptotic and Order Relations [8]
  5 Asymptotic Solution of Transcendental Equations: Real Variables [11]
  6 Asymptotic Solution of Transcendental Equations : Complex Variables [14]
  7 Definition and Fundamental Properties of Asymptotic Expansions [16]
  8 Operations with Asymptotic Expansions [19]
  9 Functions Having Prescribed Asymptotic Expansions [22]
  10 Generalizations of Poincare's Definition [24]
  11 Error Analysis; Variational Operator [27]
    Historical Notes and Additional References [29]
2 Introduction to Special Functions
  1 The Gamma Function [31]
  2 The Psi Function [39]
  3 Exponential, Logarithmic, Sine, and Cosine Integrals [40]
  4 Error Functions, Dawson's Integral, and Fresnel Integrals [43]
  5 Incomplete Gamma Functions [45]
  6 Orthogonal Polynomials [46]
  7 The Classical Orthogonal Polynomials [48]
  8 The Airy Integral [53]
  9 The Bessel Function (?) [55]
  10 The Modified Bessel Function (?) [60]
  11 The Zeta Function [61]
    Historical Notes and Additional References [64]
3 Integrals of a Real Variable
  1 Integration by Parts [66]
  2 Laplace Integrals [67]
  3 Watson's Lemma [71]
  4 The Riemann-Lebesgue Lemma [73]
  5 Fourier Integrals [75]
  6 Examples; Cases of Failure [76]
  7 Laplace's Method [80]
  8 Asymptotic Expansions by Laplace's Method; Gamma Function of Large Argument [85]
  9 Error Bounds for Watson's Lemma and Laplace's Method [89]
  10 Examples [92]
  11 The Method of Stationary Phase [96]
  12 Preliminary Lemmas [98]
  13 Asymptotic Nature of the Stationary Phase Approximation [100]
  14 Asymptotic Expansions by the Method of Stationary Phase [104]
    Historical Notes and Additional References [104]
4 Contour Integrals
  1 Laplace Integrals with a Complex Parameter [106]
  2 Incomplete Gamma Functions of Complex Argument [109]
  3 Watson's Lemma [112]
  4 Airy Integral of Complex Argument; Compound Asymptotic Expansions [116]
  5 Ratio of Two Gamma Functions; Watson's Lemma for Loop Integrals [118]
  6 Laplace's Method for Contour Integrals [121]
  7 Saddle Points [125]
  8 Examples [127]
  9 Bessel Functions of Large Argument and Order [130]
  10 Error Bounds for Laplace's Method; the Method of Steepest Descents [135]
    Historical Notes and Additional References [137]
5 Differential Equations with Regular Singularities; Hypergeometric and Legendre Functions
  1 Existence Theorems for Linear Differential Equations: Real Variables [139]
  2 Equations Containing a Real or Complex Parameter [143]
  3 Existence Theorems for Linear Differential Equations: Complex Variables [145]
  4 Classification of Singularities; Nature of the Solutions in the Neighborhood of a Regular Singularity [148]
  5 Second Solution When the Exponents Differ by an Integer or Zero [150]
  6 Large Values of the Independent Variable [153]
  7 Numerically Satisfactory Solutions [154]
  8 The Hypergeometric Equation [156]
  9 The Hypergeometric Function [159]
  10 Other Solutions of the Hypergeometric Equation [163]
  11 Generalized Hypergeometric Functions [168]
  12 The Associated Legendre Equation [169]
  13 Legendre Functions of General Degree and Order [174]
  14 Legendre Functions of Integer Degree and Order [180]
  15 Ferrers Functions [185]
    Historical Notes and Additional References [189]
6 The Liouville-Green Approximation
  1 The Liouville Transformation [190]
  2 Error Bounds: Real Variables [193]
  3 Asymptotic Properties with Respect to the Independent Variable [197]
  4 Convergence of (?) (F) at a Singularity [200]
  5 Asymptotic Properties with Respect to Parameters [203]
  6 Example: Parabolic Cylinder Functions of Large Order [206]
  7 A Special Extension [208]
  8 Zeros [211]
  9 Eigenvalue Problems [214]
  10 Theorems on Singular Integral Equations [217]
  11 Error Bounds: Complex Variables [220]
  12 Asymptotic Properties for Complex Variables [223]
  13 Choice of Progressive Paths [224]
    Historical Notes and Additional References [228]
7 Differential Equations with Irregular Singularities; Bessel and Confluent Hypergeometric Functions
  1 Formal Series Solutions [229]
  2 Asymptotic Nature of the Formal Series [232]
  3 Equations Containing a Parameter [236]
  4 Hankel Functions; Stokes' Phenomenon [237]
  5 The Function (?) [241]
  6 Zeros of (?) [244]
  7 Zeros of (?) and Other Cylinder Functions [248]
  8 Modified Bessel Functions [250]
  9 Confluent Hypergeometric Equation [254]
  10 Asymptotic Solutions of the Confluent Hypergeometric Equation [256]
  11 Whittaker Functions [260]
  12 Error Bounds for the Asymptotic Solutions in the General Case [262]
  13 Error Bounds for Hankel's Expansions [266]
  14 Inhomogeneous Equations [270]
  15 Struve's Equation [274]
    Historical Notes and Additional References [277]
8 Sums and Sequences
  1 The Euler-Maclaurin Formula and Bernoulli's Polynomials [279]
  2 Applications [284]
  3 Contour Integral for the Remainder Term [289]
  4 Stirling's Series for In Г(z) [293]
  5 Summation by Parts [295]
  6 Barnes' Integral for the Hypergeometric Function [299]
  7 Further Examples [302]
  8 Asymptotic Expansions of Entire Functions [307]
  9 Coefficients in a Power-Series Expansion; Method of Darboux [309]
  10 Examples [311]
  11 Inverse Laplace Transforms; Haar's Method [315]
    Historical Notes and Additional References [321]
9 Integrals: Further Methods
  1 Logarithmic Singularities [322]
  2 Generalizations of Laplace's Method [325]
  3 Example from Combinatoric Theory [329]
  4 Generalizations of Laplace's Method (continued) [331]
  5 Examples [334]
  6 More General Kernels [336]
  7 Nicholson's Integral for (формула) [340]
  8 Oscillatory Kernels [342]
  9 Bleistein's Method [344]
  10 Example [346]
  11 The Method of Chester, Friedman, and Ursell [351]
  12 Anger Functions of Large Order [352]
  13 Extension of the Region of Validity [358]
    Historical Notes and Additional References [361]
10 Differential Equations with a Parameter: Expansions in Elementary Functions
  1 Classification and Preliminary Transformations [362]
  2 Case I: Formal Series Solutions [364]
  3 Error Bounds for the Formal Solutions [366]
  4 Behavior of the Coefficients at a Singularity [368]
  5 Behavior of the Coefficients at a Singularity (continued) [369]
  6 Asymptotic Properties with Respect to the Parameter [371]
  7 Modified Bessel Functions of Large Order [374]
  8 Extensions of the Regions of Validity for the Expansions of the Modified Bessel Functions [378]
  9 More General Forms of Differential Equation [382]
  10 Inhomogeneous Equations [386]
  11 Example: An Inhomogeneous Form of the Modified Bessel Equation [388]
    Historical Notes and Additional References [391]
11 Differential Equations with a Parameter: Turning Points
  1 Airy Functions of Real Argument [392]
  2 Auxiliary Functions for Real Variables [394]
  3 The First Approximation [397]
  4 Asymptotic Properties of the Approximation; Whittaker Functions with m Large [401]
  5 Real Zeros of the Airy Functions [403]
  6 Zeros of the First Approximation [405]
  7 Higher Approximations [408]
  8 Airy Functions of Complex Argument [413]
  9 Asymptotic Approximations for Complex Variables [416]
  10 Bessel Functions of Large Order [419]
  11 More General Form of Differential Equation [426]
  12 Inhomogeneous Equations [429]
    Historical Notes and Additional References [433]
12 Differential Equations with a Parameter: Simple Poles and Other Transition Points
  1 Bessel Functions and Modified Bessel Functions of Real Order and Argument [435]
  2 Case III: Formal Series Solutions [438]
  3 Error Bounds: Positive (?) [440]
  4 Error Bounds: Negative (?) [443]
  5 Asymptotic Properties of the Expansions [447]
  6 Determination of Phase Shift [449]
  7 Zeros [451]
  8 Auxiliary Functions for Complex Arguments [453]
  9 Error Bounds: Complex и and f [457]
  10 Asymptotic Properties for Complex Variables [460]
  11 Behavior of the Coefficients at Infinity [462]
  12 Legendre Functions of Large Degree: Real Arguments [463]
  13 Legendre Functions of Large Degree: Complex Arguments [470]
  14 Other Types of Transition Points [474]
    Historical Notes and Additional References [478]
13 Connection Formulas for Solutions of Differential Equations
  1 Introduction [480]
  2 Connection Formulas at a Singularity [480]
  3 Differential Equations with a Parameter [482]
  4 Connection Formula for Case III [483]
  5 Application to Simple Poles [487]
  6 Example: The Associated Legendre Equation [490]
  7 The Cans-Jeffreys Formulas: Real-Variable Method [491]
  8 Two Turning Points [494]
  9 Bound States [497]
  10 Wave Penetration through a Barrier. I [501]
  11 Fundamental Connection Formula for a Simple Turning Point in the Complex Plane [503]
  12 Example: Airy's Equation [507]
  13 Choice of Progressive Paths [508]
  14 The Cans-Jeffreys Formulas: Complex-Variable Method [510]
  15 Wave Penetration through a Barrier. II [513]
    Historical Notes and Additional References [516]
14 Estimation of Remainder Terms
  1 Numerical Use of Asymptotic Approximations [519]
  2 Converging Factors [522]
  3 Exponential Integral [523]
  4 Exponential Integral (continued) [527]
  5 Confluent Hypergeometric Function [531]
  6 Euler's Transformation [536]
  7 Application to Asymptotic Expansions [540]
    Historical Notes and Additional References [543]
Answers to Exercises [545]
References [548]
Index of Symbols [561]
General Index [563]
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