The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation

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The Stokes phenomenon refers to the emergence of jump discontinuities in asymptotic expansions at specific rays in the complex plane. This book presents a radical theory for the phenomenon by ...
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Dingle’s Rules and Theory of Terminants

Pp. 13-21 (9)

Victor Kowalenko

Abstract

Dingle’s rules for deriving the changing forms of an asymptotic expansion as a result of the Stokes phenomenon are introduced. Of these rules only Nos. 1, 7 and 8 are necessary for the analysis in later chapters. Then with the aid of these rules his theory of terminants is presented, which is important because the asymptotic expansions of a multitude of mathematical functions can be approximated for large values of the summation index by these divergent series whose coefficients possess gamma function growth. There are basically two types of terminants, each possessing different properties according to Dingle’s rules. Meaningful values for both types of terminants are obtained by Borel summation, which is shown to be a method of regularising asymptotic series.

Affiliation:

Melbourne, Australia