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Algebraic analysis of singular perturbation theory. Transl. from the Japanese by Goro Kato. (English) Zbl 1100.34004

Translations of Mathematical Monographs 227. Iwanami Series in Modern Mathematics. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3547-5/pbk). xiii, 128 p. (2005).
The main theme of the book is singular perturbation theory for ordinary differential equations, in particular the so-called WKB method. In Chapter 1, a brief introduction to Borel summation of formal power series, along with some interesting examples, is given, followed by WKB analysis of Schrödinger equations in Chapter 2. The remaining two chapters are devoted to applications of this analysis, first to global problems for ODEs, such as the monodromy group of a Fuchsian equation, and then to Painlevé transcendents. A final section contains an outline of future directions of research in this area.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
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