Introduction to Asymptotics and Special Functions

PrefacePreface to Asymptotics and Special Functions1 Introduction to Asymptotic Analysis 1 Origin of Asymptotic Expansions 2 The Symbols ~, o, and O 3 The Symbols ~, o, and O (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 ReferencesAnswers to ExercisesReferencesIndex of SymbolGeneral Index