By Takahiro Kawai and Yoshitsugu Takei

ISBN-10: 0821835475

ISBN-13: 9780821835470

The subject of this ebook is the learn of singular perturbations of standard differential equations, i.e., perturbations that characterize suggestions as asymptotic sequence instead of as analytic capabilities in a perturbation parameter. the most strategy used is the so-called WKB (Wentzel-Kramers-Brillouin) strategy, initially invented for the learn of quantum-mechanical structures. The authors describe intimately the WKB technique and its functions to the examine of monodromy difficulties for Fuchsian differential equations and to the research of Painleve capabilities. the amount is acceptable for graduate scholars and researchers drawn to differential equations and distinctive services.

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**Example text**

2p -h 1) can never be connected to any other turning point or aj itself by a Stokes curve. 21) defining the Borel sum. , with the assistance of a computer. For the computation of the monodromy group, choose a base point xq (avoiding a turning point and points on Stokes curves). 9) = exp y/ ^odd i / d Xq Sodd 48 3. 9) are defined in terms of the integral, the integration path needs to be specified. 5 itself). Particularly in the case of Fuchsian type differential equations, the choice of the integration path is crucially important as there are several regular singular points besides the point at infin ity.

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.

Hence every Xj is holomorphic on a fixed neighborhood independent of j. n) must be 0. For an odd integer n, each term of f n must have at least one of either Xj {j: odd, j < n) or its derivative as a factor. Since x\ is identically 0 , by induction, we conclude that all the Xj {j: odd) are 0. 36 2. 50), have been proved. 71), we may consider t = x q {x ) in a neighborhood of X = 0 as a new coordinate function. 76). 80) du < sup l/(i)| + sup \u{t)\ dt It|

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

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