By Takahiro Kawai and Yoshitsugu Takei

The subject of this publication is the research of singular perturbations of normal differential equations, i.e., perturbations that signify recommendations as asymptotic sequence instead of as analytic features in a perturbation parameter. the most strategy used is the so-called WKB (Wentzel-Kramers-Brillouin) process, initially invented for the learn of quantum-mechanical platforms. The authors describe intimately the WKB approach and its functions 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 attracted to differential equations and certain features.

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**Example text**

Assume that all the turning points of ( 2. 94) the turning points are never connected by Stokes curves. 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).

17. 15. 15 cannot handle the constant multiple between and 'ip±. 11)). 15. 15 and the transformation of a W K B solution are two different aspects of the same thing. 15. 15. 3. CONNECTION FORMULA— THE GENERAL CASE Let X = . 51) as the equations for {xj}j>o to satisfy. 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.

3)' d egF = 2^ -h 2 and degG = -h 3 {g e Z , g > - 1), then X = 00 is not a singular point. 1) are only the zeros of G (x). For a differential equation of Fuchsian type of this form, our argument also applies. 1. MONODROMY GROUP; FUCHSIAN TYPE 45 Choose a base point xq in P^(C) \ {bo, . . 1) around xq. 1) is linear, an arbitrary solution does not have any other singular point besides { 60, . . , 6^+ 2}. There fore, we may consider the analytic continuation of ('0 i ,' 02) along any (C) \ { 60, •••, ^p+2} having xq as the initial point.