By Takahiro Kawai and Yoshitsugu Takei

ISBN-10: 0821835475

ISBN-13: 9780821835470

The subject of this publication is the examine of singular perturbations of standard differential equations, i.e., perturbations that symbolize suggestions as asymptotic sequence instead of as analytic services in a perturbation parameter. the most procedure used is the so-called WKB (Wentzel-Kramers-Brillouin) strategy, initially invented for the examine of quantum-mechanical platforms. 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 features. the amount is appropriate for graduate scholars and researchers attracted to differential equations and unique services.

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**Sample text**

95) = ^ — exp ( ± [ V *^ o d d \ Sodd dx J a over each region Uj. b) is for the case: ^/Q{x) dx > 0. The signs ± correspond to the counter-clockwise and clockwise crossing of T of the path of analytic continuation from U\ to U2 with the turning point as the reference point. 1 The W K B method is used to find a formal power series solu tion of the Schrödinger equation (— rj^Q{x))'ip{x,rj) = 0 (for a large parameter rj). 2 Even though the W K B solution is a divergent series, it works well with the Borel sum.

1) 2xqx[ + XqXi = 0, 34 2. n) are linear differential equations having x = 0 as a singular point. Note that this singular point is a regular singular point. Because of this reg ular singular character, Xj (x) is recursively and uniquely determined once xo(x) is determined. This fact shows that our transformation and the singular perturbation theory are two aspects of an entity. This is a conceptually important observation. ) Actually we will encounter a similar situation (although technically much more difficult) in discussing the transformation of the underly ing Schrödinger equation to construct an appropriate transformation of the Painleve transcendents.

17) (A is a turning point and O is a regular singular point). 7. 12). ) We fix the numbering of turning points and regular singular points as follows: ao = —3, a\ = —1 /3 , U2 = 1 /3 , as = 3, { bo = exp(33z7r/24), bi = exp(z7r/24), 62 = exp(1727r/24), 63 = 00. 17) is an example of the case g = 1. 1. 2. 9). 2, the Stokes curves are drawn with thin solid lines. For the com putation of the monodromy group, only the topological configuration of Stokes curves is relevant. 2. 18) 91co < 0, fHci > 0, fHc2 > 0, fHcs < 0.

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

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