By Takahiro Kawai and Yoshitsugu Takei

ISBN-10: 0821835475

ISBN-13: 9780821835470

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

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**Extra info for Algebraic Analysis of Singular Perturbation Theory**

**Sample text**

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. 24) j S2j-\{x)dx S2j-i{x )d x , where x is the point on a sheet different from the one containing x, of the Riemann surface of {x — or more precisely of \/Q{x). 24) is shown in Figure 2. 1: the sign (—1) appears for the integral along the dotted line which is considered to be on the second sheet. Furthermore, the orientation of the integration path along the dotted line is opposite to that along the solid line.

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. There fore, it is an important question to ask how the analytically continued W K B solution to a neighboring region is related to the W K B solution on that region. The connection formula by Voros in [65] answers this question. The answer for this question is given in the previous section where Q = x. 15) in a neighborhood of a simple turning point.

A connected open set) which is bounded by Stokes curves is said to be a Stokes region. 11. W e will show several examples of Stokes regions for simple potentials Q{x). 7 of Chapter 3. Those examples are illustrated by computer-aided numerical computation. Contact Prof. Takashi Aoki for the computer programming at Kinki Univer sity, Department of Mathematics, Higashi-Osaka, 577-8502 Japan. Although we do not give the proof here but refer to Voros [65], the following fact is important. 12. 4 never happens).

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

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