By Takahiro Kawai and Yoshitsugu Takei

The subject of this ebook is the research of singular perturbations of normal differential equations, i.e., perturbations that characterize ideas as asymptotic sequence instead of as analytic features in a perturbation parameter. the most process used is the so-called WKB (Wentzel-Kramers-Brillouin) strategy, initially invented for the examine of quantum-mechanical structures. The authors describe intimately the WKB approach and its purposes to the examine of monodromy difficulties for Fuchsian differential equations and to the research of Painleve capabilities. the amount is appropriate for graduate scholars and researchers drawn to differential equations and distinct capabilities.

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**Example text**

2. 12) /c = 0 , . . , ^ + 2. 8), is again concerned with the fixed singular ity of the Borel transform of the W K B solution. ) W e then obtain the following result on the behavior at X = bk oi the odd part 5odd(^>^) oi the logarithmic derivative of the W K B solution. 6. 13) R^^5odd(x,7,) = c J l + ^ . P r o o f . 6), it is clear that each Sj{x) has a pole at x = 5^ of order at most 1. 1. 15) S{x,T])= ^ Then we have fj,oc{x-hk)°‘'q~^. 3). 13). 13) holds including the choice of the branch.

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.

1). 4. 1) does not contain the term of first order, its monodromy representation is a mapping into 5 L ( 2 ,C ) . ) The image (more precisely, its conjugate class) of the monodromy representation forms the monodromy group. We will observe how explicitly the monodromy group (or the mon odromy representation) can be computed. For example, for the case ^ = —1, assume that x = 60 is the only regular singular point besides the point at infinity. 1) is then given by (x — ho)^, where a is the characteristic exponent at x = 60, and hence we can compute the monodromy group easily.