Temme N.M. Asymptotic Methods for Integrals

World Scientific, 2015. — 587 p. — (Series in Analysis 06). — ISBN13: 9781322501130.This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals.Basic Methods for IntegralsExpansions of Laplace-type integrals: Watson¡¯s lemma
The method of Laplace
The saddle point method and paths of steepest descent
The Stokes phenomenon
Basic Methods: Examples for Special Functions
The gamma function
Incomplete gamma functions
The Airy functions
Bessel functions: Large argument
Kummer functions
Parabolic cylinder functions: Large argument
The Gauss hypergeometric function
Examples of 3F2-polynomials
Other Methods for Integrals
The method of stationary phase
Coefficients of a power series. Darboux¡¯s method
Mellin.Barnes integrals and Mellin convolution integrals
Alternative expansions of Laplace-type integrals
Two-point Taylor expansions
Hermite polynomials as limits of other classical orthogonal polynomials
Uniform Methods for Integrals
An overview of standard forms
A saddle point near a pole
Saddle point near algebraic singularity
Two coalescing saddle points: Airy-type expansions
Hermite-type expansions of integrals
The vanishing saddle point
A moving endpoint: Incomplete Laplace integrals
An essential singularity: Bessel-type expansions
Expansions in terms of Kummer functions
Uniform Examples for Special Functions
Legendre functions
Parabolic cylinder functions: Large parameter
Coulomb wave functions
Laguerre polynomials: Uniform expansions
Generalized Bessel polynomials
Stirling numbers 4
Asymptotics of the integral
A Class of Cumulative Distribution Functions
Expansions of a class of cumulative distribution functions
Incomplete gamma functions: Uniform
Incomplete beta function
Non-central chi-square, Marcum functions
A weighted sum of exponentials
A generalized incomplete gamma function
Asymptotic inversion of cumulative distribution functions