By Takahiro Kawai and Yoshitsugu Takei

ISBN-10: 0821835475

ISBN-13: 9780821835470

The subject of this booklet is the research of singular perturbations of normal differential equations, i.e., perturbations that symbolize strategies 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 approach and its purposes to the research of monodromy difficulties for Fuchsian differential equations and to the research of Painleve services. the amount is acceptable for graduate scholars and researchers attracted to differential equations and distinctive services.

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

**Sample text**

G. in p. 512; it was published in 1978), and it was in those days when Professor Silverstone wrote [57]. We really admire his precise and tenacious mind for having argued so sharply and clearly against the tide at that time. The paper [57] is a role model for articles in exact science; we strongly recommend the reader to read [57]. 14. The ‘theory of transformations’ will play a fun damental role also in the W K B analysis of the Painleve transcendent in Chapter 4. 15. Let x be the independent variable in ( 2.

5 above is reduced to the connection formula for the classical hypergeometric function, through the Borel transformation. 5(i) above might look like a formal relation at first sight, the Borel transformation renders it to be an exact relation expressed in terms of microdiiferential operators acting on analytic functions. CHAPTER 3 Applications of W K B Analysis to Global Problems As we mentioned in ‘Summary and Overview’ , the connection formula for a W K B solution studied in Chapter 2 is quite effective for the global problems of differential equations.

21). ) Note that '^ ± s(x) ^ ± , b (5, y) stands for the discontinuity along {{x,y) e W ;3 y 3{±s{x)), 3Ky > 9 1 (± s (x ))}, where ± signs are assigned in this order. 2. 21. 85) A _s(ä )i^ _,ß (x ,y ) = i'ip+^B{x,y). P ro o f . 18, we will show that our claim can be reduced to the explicit result for the case Q = x. The the oretical aspect core of our reasoning is the following: multiplication by the formal power series f{x,r]) = J2j>N without expo nential terms acts as a microdifferential operator on Borel transforms (see Kashiwara-Kawai-Kimura [34, Chap.

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

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