From Divergent Power Series to Analytic Functions

Abstract

I am grateful to Y. Sibuya and B.L.J. Braaksma, who introduced me to the beautiful theory of multisummability in a joint seminar at the University of Minneapolis, in Spring of 1990, and I pray that they will consider me a good student. I also owe thanks to my student Andreas Beck, who went over the proofs and did all exercises, and to Sabine Lebhart for carefully typing the manuscript. Finally, I wish to apologize to my wife Christel, for spending parts of our vacation on a Dutch island on writing this text.

On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations

“…As a preparation for the proof of the main theorems, following [1], we review the definition and some fundamental properties for the multisummability in this section. We basically employ the same notation as in [1] except that we use a large parameter η here instead of a small parameter = η −1 as an asymptotic parameter.…”

“…We basically employ the same notation as in [1] except that we use a large parameter η here instead of a small parameter = η −1 as an asymptotic parameter.…”

“…Proposition 3.5 is proved by using the so-called Cauchy-Heine transform and Proposition 3.4. For the Cauchy-Heine transform see [1,Chap. 4].…”

On k-summability of formal solutions for certain partial differential operators with polynomial coefficients

“…Note that this formula is same with that in [1,Sec. 5.3] although the expression is a little different from it.…”

A Generalisation of the Cauchy–Kovalevskaïa Theorem

“…Borel summability was extended to arbitrary Gevrey series by (Écalle 1981 andRamis 1980). Other recent references are Balser's textbook and Malgrange's lectures on divergent power series (Balser 1994;Malgrange 1995).…”