By Takahiro Kawai and Yoshitsugu Takei

The subject of this e-book is the learn 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) technique, initially invented for the research of quantum-mechanical structures. The authors describe intimately the WKB technique and its functions to the research of monodromy difficulties for Fuchsian differential equations and to the research of Painleve capabilities. the amount is appropriate for graduate scholars and researchers attracted to differential equations and detailed capabilities.

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**Example text**

Then the Borel sum of a WKB solution is well defined on each Stokes region, and it determines a holomorphic function of x there. Since a W K B solution satisfies the linear differential equation (2. 1), the Borel sum of the W K B solution can be analytically contin ued beyond each Stokes curve. 2) and also at the end of Section 2. 2, the integration path of the integral defining the Borel sum meets another singular point. Consequently, we need to change the integration path for the integral beyond the Stokes curve.

X,y)-space. (See Sato-Kawai-Kashiwara [55, Chap. II] and Kashiwara-Kawai-Kimura [34, Chap. 23) = m ), dt dyjt) dt dm dt dm = 0, dt - Q i x i t ) ) ^ ^ = 0, (Ç, v) ¥= 0. 1. 1. For S2j-i{x ) dx (the wavy line is a cut determining the branch of the dotted line is the integration path on the second sheet, and X on the second sheet corresponds to x). Recall that S2j - i consists of only the odd power terms of y/Q{x). Therefore, in a neighborhood of the simple turning point a, S2j - i shows the singularity of the form {x — where I is odd.

1 . 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.