Asymptotics and Special Functions

Preface1 Introduction to Asymptotic Analysis 1 Origin of Asymptotic Expansions 2 The Symbols ~, o, and Ο 3 The Symbols ~, o, and Ο (Continued) 4 Integration and Differentiation of Asymptotic and Order Relations 5 Asymptotic Solution of Transcendental Equations: Real Variables 6 Asymptotic Solution of Transcendental Equations: Complex Variables 7 Definition and Fundamental Properties of Asymptotic Expansions 8 Operations with Asymptotic Expansions 9 Functions Having Prescribed Asymptotic Expansions 10 Generalizations of Poincaré's Definition 11 Error Analysis; Variational Operator Historical Notes and Additional References2 Introduction to Special Functions 1 The Gamma Function 2 The Psi Function 3 Exponential, Logarithmic, Sine, and Cosine Integrals 4 Error Functions, Dawson's Integral, and Fresnel Integrals 5 Incomplete Gamma Functions 6 Orthogonal Polynomials 7 The Classical Orthogonal Polynomials 8 The Airy Integral 9 The Bessel Function Jv(z) 10 The Modified Bessel Function Iv(z) 11 The Zeta Function Historical Notes and Additional References3 Integrals of a Real Variable 1 Integration by Parts 2 Laplace Integrals 3 Watson's Lemma 4 The Riemann-Lebesgue Lemma 5 Fourier Integrals 6 Examples; Cases of Failure 7 Laplace's Method 8 Asymptotic Expansions by Laplace's Method; Gamma Function of Large Argument 9 Error Bounds for Watson's Lemma and Laplace's Method 10 Examples 11 The Method of Stationary Phase 12 Preliminary Lemmas 13 Asymptotic Nature of the Stationary Phase Approximation 14 Asymptotic Expansions by the Method of Stationary Phase Historical Notes and Additional References4 Contour Integrals 1 Laplace Integrals with a Complex Parameter 2 Incomplete Gamma Functions of Complex Argument 3 Watson's Lemma 4 Airy Integral of Complex Argument; Compound Asymptotic Expansions 5 Ratio of Two Gamma Functions; Watson's Lemma for Loop Integrals 6 Laplace's Method for Contour Integrals 7 Saddle Points 8 Examples 9 Bessel Functions of Large Argument and Order 10 Error Bounds for Laplace's Method; The Method of Steepest Descents Historical Notes and Additional References5 Differential Equations with Regular Singularities; Hypergeometric and Legendre Functions 1 Existence Theorems for Linear Differential Equations: Real Variables 2 Equations Containing a Real or Complex Parameter 3 Existence Theorems for Linear Differential Equations: Complex Variables 4 Classification of Singularities; Nature of the Solutions in the Neighborhood of a Regular Singularity 5 Second Solution When the Exponents Differ by an Integer or Zero 6 Large Values of the Independent Variable 7 Numerically Satisfactory Solutions 8 The Hypergeometric Equation 9 The Hypergeometric Function 10 Other Solutions of the Hypergeometric Equation 11 Generalized Hypergeometric Functions 12 The Associated Legendre Equation 13 Legendre Functions of General Degree and Order 14 Legendre Functions of Integer Degree and Order 15 Ferrers Functions Historical Notes and Additional References6 The Liouville-Green Approximation 1 The Liouville Transformation 2 Error Bounds: Real Variables 3 Asymptotic Properties with Respect to the Independent Variable 4 Convergence of V (F) at a Singularity 5 Asymptotic Properties with Respect to Parameters 6 Example: Parabolic Cylinder Functions of Large Order 7 A Special Extension 8 Zeros 9 Eigenvalue Problems 10 Theorems on Singular Integral Equations 11 Error Bounds: Complex Variables 12 Asymptotic Properties for Complex Variables 13 Choice of Progressive Paths Historical Notes and Additional References7 Differential Equations with Irregular Singularities; Bessel and Confluent Hypergeometric Functions 1 Formal Series Solutions 2 Asymptotic Nature of the Formal Series 3 Equations Containing a Parameter 4 Hankel Functions; Stokes' Phenomenon 5 The Function Yv(z) 6 Zeros of Jv(z) 7 Zeros of Yv(z) and Other Cylinder Functions 8 Modified Bessel Functions 9 Confluent Hypergeometric Equation 10 Asymptotic Solutions of the Confluent Hypergeometric Equation 11 Whittaker Functions 12 Error Bounds for the Asymptotic Solutions in the General Case 13 Error Bounds for Hankel's Expansions 14 Inhomogeneous Equations 15 Struve's Equation Historical Notes and Additional References8 Sums and Sequences 1 The Euler-Maclaurin Formula and Bernoulli's Polynomials 2 Applications 3 Contour Integral for the Remainder Term 4 Stirling's Series for In Γ(z) 5 Summation by Parts 6 Barnes' Integral for the Hypergeometric Function 7 Further Examples 8 Asymptotic Expansions of Entire Functions 9 Coefficients in a Power-Series Expansion; Method of Darboux 10 Examples 11 Inverse Laplace Transforms; Haar's Method Historical Notes and Additional References9 Integrals: Further Methods 1 Logarithmic Singularities 2 Generalizations of Laplace's Method 3 Example from Combinatoric Theory 4 Generalizations of Laplace's Method (Continued) 5 Examples 6 More General Kernels 7 Nicholson's Integral for J2v(z) + Y2v(z) 8 Oscillatory Kernels 9 Bleistein's Method 10 Example 11 The Method of Chester, Friedman, and Ursell 12 Anger Functions of Large Order 13 Extension of the Region of Validity Historical Notes and Additional References10 Differential Equations with a Parameter: Expansions in Elementary Functions 1 Classification and Preliminary Transformations 2 Case I: Formal Series Solutions 3 Error Bounds for the Formal Solutions 4 Behavior of the Coefficients at a Singularity 5 Behavior of the Coefficients at a Singularity (Continued) 6 Asymptotic Properties with Respect to the Parameter 7 Modified Bessel Functions of Large Order 8 Extensions of the Regions of Validity for the Expansions of the Modified Bessel Functions 9 More General Forms of Differential Equation 10 Inhomogeneous Equations 11 Example: An Inhomogeneous Form of the Modified Bessel Equation Historical Notes and Additional References11 Differential Equations with a Parameter: Turning Points 1 Airy Functions of Real Argument 2 Auxiliary Functions for Real Variables 3 The First Approximation 4 Asymptotic Properties of the Approximation; Whittaker Functions with m Large 5 Real Zeros of the Airy Functions 6 Zeros of the First Approximation 7 Higher Approximations 8 Airy Functions of Complex Argument 9 Asymptotic Approximations for Complex Variables 10 Bessel Functions of Large Order 11 More General Form of Differential Equation 12 Inhomogeneous Equations Historical Notes and Additional References12 Differential Equations with a Parameter: Simple Poles and Other Transition Points 1 Bessel Functions and Modified Bessel Functions of Real Order and Argument 2 Case III: Formal Series Solutions 3 Error Bounds: Positive ζ 4 Error Bounds: Negative ζ 5 Asymptotic Properties of the Expansions 6 Determination of Phase Shift 7 Zeros 8 Auxiliary Functions for Complex Arguments 9 Error Bounds: Complex u and ζ 10 Asymptotic Properties for Complex Variables 11 Behavior of the Coefficients at Infinity 12 Legendre Functions of Large Degree: Real Arguments 13 Legendre Functions of Large Degree: Complex Arguments 14 Other Types of Transition Points Historical Notes and Additional References13 Connection Formulas for Solutions of Differential Equations 1 Introduction 2 Connection Formulas at a Singularity 3 Differential Equations with a Parameter 4 Connection Formula for Case III 5 Application to Simple Poles 6 Example: The Associated Legendre Equation 7 The Gans-Jeffreys Formulas: Real-Variable Method 8 Two Turning Points 9 Bound States 10 Wave Penetration Through a Barrier. I 11 Fundamental Connection Formula for a Simple Turning Point in the Complex Plane 12 Example: Airy's Equation 13 Choice of Progressive Paths 14 The Gans-Jeffreys Formulas: Complex-Variable Method 15 Wave Penetration through a Barrier. II Historical Notes and Additional References14 Estimation of Remainder Terms 1 Numerical Use of Asymptotic Approximations 2 Converging Factors 3 Exponential Integral 4 Exponential Integral (Continued) 5 Confluent Hypergeometric Function 6 Euler's Transformation 7 Application to Asymptotic Expansions Historical Notes and Additional ReferencesAnswers to ExercisesReferencesIndex of SymbolsGeneral Index