Introduction to Asymptotics and Special Functions
Free Global Shipping
No minimum orderDescription
Introduction to Asymptotics and Special Functions is a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable are discussed, along with contour integrals and differential equations with regular and irregular singularities. The Liouville-Green approximation is also considered. Comprised of seven chapters, this volume begins with an overview of the basic concepts and definitions of asymptotic analysis and special functions, followed by a discussion on integrals of a real variable. Contour integrals are then examined, paying particular attention to Laplace integrals with a complex parameter and Bessel functions of large argument and order. Subsequent chapters focus on differential equations having regular and irregular singularities, with emphasis on Legendre functions as well as Bessel and confluent hypergeometric functions. A chapter devoted to the Liouville-Green approximation tackles asymptotic properties with respect to parameters and to the independent variable, eigenvalue problems, and theorems on singular integral equations. This monograph is intended for students needing only an introductory course to asymptotics and special functions.
Table of Contents
Preface
Preface to Asymptotics and Special Functions
1 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 References
2 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 References
3 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 References
4 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 References
5 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 References
6 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 References
7 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 References
Answers to Exercises
References
Index of Symbol
General Index
Product details
- No. of pages: 312
- Language: English
- Copyright: © Academic Press 1974
- Published: March 28, 1974
- Imprint: Academic Press
- eBook ISBN: 9781483267081
About the Author
F. W. J. Olver
Ratings and Reviews
There are currently no reviews for "Introduction to Asymptotics and Special Functions"