By Takahiro Kawai and Yoshitsugu Takei

ISBN-10: 0821835475

ISBN-13: 9780821835470

The subject of this ebook is the research of singular perturbations of standard differential equations, i.e., perturbations that signify options as asymptotic sequence instead of as analytic features in a perturbation parameter. the most process used is the so-called WKB (Wentzel-Kramers-Brillouin) procedure, initially invented for the research of quantum-mechanical structures. The authors describe intimately the WKB procedure and its purposes to the learn of monodromy difficulties for Fuchsian differential equations and to the research of Painleve features. the quantity is acceptable for graduate scholars and researchers drawn to differential equations and certain services.

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**Sample text**

18) and (ii) the explicit analysis (Section 2. /dx^‘ + rj‘^x)'ip{x,rj) = 0 , the canonical equation in the trans formation theory. 5 above is reduced to the connection formula for the classical hypergeometric function, through the Borel transformation. 5(i) above might look like a formal relation at first sight, the Borel transformation renders it to be an exact relation expressed in terms of microdiiferential operators acting on analytic functions. CHAPTER 3 Applications of W K B Analysis to Global Problems As we mentioned in ‘Summary and Overview’ , the connection formula for a W K B solution studied in Chapter 2 is quite effective for the global problems of differential equations.

85). 22. It is more natural from the viewpoint of the gen eral theory of differential equations to interpret the above discussions 40 2. 93) can be transformed by microdifferential operators or not. Note, how ever, even if the principal parts of L and M are the same through an appropriate coordinate transformation x = xq{x ), L and M cannot be related by an inner automorphism, that is, we cannot find A such that A~^MA = L. See, for example, Aoki-Yoshida [11]. Before we begin a new chapter, we will rewrite the connection formulae for W K B solutions into general and manageable forms.

29) Cj is a constant, which can be proved by induction. 32) x -i/4 “ r (l/2 ) / 23/2 3^ only. Consequently, - 1/2 | l + l . f x - » / " ( ä , ± | i > '" ) ( y ± | x V ")' + • + 22 2. 20). 34) 1 (Ph± dt^ - A 0. 35) which is the hypergeometric differential equation [46, p. 33), /i+ can be written as g+{s)/y/s in a neighborhood of s = 0 , and h - can be written as g -{s — l ) /\ /s — 1 in a neighborhood of s = 1. ) A system of fundamental solutions of a hypergeometric differential equation is given by the celebrated Rummer’s relations (see [46, pp.

### Algebraic Analysis of Singular Perturbation Theory by Takahiro Kawai and Yoshitsugu Takei

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