Borel-Laplace transform and asymptotic theory : introduction to resurgent analysis
著者
書誌事項
Borel-Laplace transform and asymptotic theory : introduction to resurgent analysis
CRC Press, c1996
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注記
Includes bibliographical references and index
内容説明・目次
内容説明
The resurgent function theory introduced by J. Ecalle is one of the most interesting theories in mathematical analysis. In essence, the theory provides a resummation method for divergent power series (e.g., asymptotic series), and allows this method to be applied to mathematical problems. This new book introduces the methods and ideas inherent in resurgent analysis. The discussions are clear and precise, and the authors assume no previous knowledge of the subject. With this new book, mathematicians and other scientists can acquaint themselves with an interesting and powerful branch of asymptotic theory - the resurgent functions theory - and will learn techniques for applying it to solve problems in mathematics and mathematical sciences.
目次
- Part 1 Introduction - resurgent analysis in the theory of differential equations: singular points of ordinary differential equations - classification of singular points, the Borel-Laplace transform, the Euler example
- equations on an infinite cylinder - modification of the Borel-Laplace transform, asymptotics of functions of exponential growth, asymptotic expansions of solutions
- semi-classical approximations - WKB-expansions (elementary theory), exact WKB-approximation (quantum oscillator), asymptotics at infinity (the airy equation). Part 2 Borel-Laplace transform: enitre functions of exponential type - definitions, the Borel-Laplace transform, examples
- hyperfunctions with compact support - definitions, the Borel-Laplace transform, examples
- hyperfunctions of exponential growth - definitions, the Borel-Laplace transform, generalized hyperfunctions, examples
- microfunctions - endlessly continuable hyperfunctions, microfunctions and their Borel-Laplace transform, microlocalization, examples. Part 3 Resurgent analysis: preliminary remarks
- resurgent functions - definition of resurgent functions, the connection homomorphism and the Stokes phenomenon, asymptotic expansions, resurgent functions with simple singularities, generalizations of the notion of a resurgent function - resurgent representation
- investigation near focal points - Legendre uniformization
- investigation near focal points - connection homomorphism - the monodromy properties of a resurgent function, alient derivatives, alient differential equations, Stokes phenomenon and univaluedness
- examples - resurgent functions of the airy type, special functions of higher order, cylinder-parabolic functions. Part 4 Applications: ordinary differential equations - reduction to the Volterra equation, analytic continuation of terms of the Neumann series, convergence of the Neumann series, resurgent solutions to ordinary differential equations
- partial differential equations - asymptotic solutions to the Schrodinger equation, general equations with polynomial coefficients, examples
- the Saddle Point method - statement of the problem, one-dimensional case, multi-dimensional case. Appendix - integral transforms of ramifying analytic functions: integral transform of homogeneous functions
- transforms, associated with the F-transform
- the R-transform
- the delta/deltas transform.
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