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

By Takahiro Kawai and Yoshitsugu Takei

The subject of this publication is the learn of singular perturbations of standard differential equations, i.e., perturbations that characterize options as asymptotic sequence instead of as analytic capabilities 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 structures. The authors describe intimately the WKB process and its functions to the learn of monodromy difficulties for Fuchsian differential equations and to the research of Painleve services. the amount is appropriate for graduate scholars and researchers drawn to differential equations and distinctive services.

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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.

23) below. , (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.

4 . Stokes curves and Stokes regions for simple potentials Q{x) (A is a turning point and © is a (regular) singular point). ‘theory of transformations’ and Borel resummation. To tell the truth, we were overwhelmed by his sharp and minute logic. As we began our study in this field by reading Bender-Wu [14] and Voros [65], we were not bothered by controversies over the legitimacy of W K B analysis. But, actually, the ambiguous traditional W K B analysis (particularly 28 2. g. in p. 512; it was published in 1978), and it was in those days when Professor Silverstone wrote [57].

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