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On an integral representation of resurgent functions. (English. Russian original) Zbl 1065.30023
Differ. Equ. 37, No. 9, 1291-1302 (2001); translation from Differ. Uravn. 37, No. 9, 1229-1238 (2001).
In this paper a detailed exposition of the asymptotic approach to the resurgent function theory is given. Asymptotic expansions for analytic functions of exponential growth are studied via a complex analytic analogous of the Laplace transform, called Borel-Laplace transform. Interesting examples and counterexamples are given. Motivated by some special effects (“the possibility of different growth exponents in different directions”) the authors sketch a generalization of the theory of Borel-Laplace transform for functions of several variables. As an application resurgent solutions with simple singularities for ordinary differential equations are developed.
MSC:
30D15 Special classes of entire functions of one complex variable and growth estimates
30E15 Asymptotic representations in the complex plane
34M37 Resurgence phenomena (MSC2000)
34Exx Asymptotic theory for ordinary differential equations
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
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