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The Borel Transform and Its Use in the Summation of Asymptotic Expansions
Author(s) -
ByattSmith J. G.
Publication year - 1999
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.1034136
Subject(s) - mathematics , asymptotic expansion , zero (linguistics) , independent and identically distributed random variables , mellin transform , mathematical analysis , real line , asymptotic formula , connection (principal bundle) , term (time) , asymptotic analysis , laplace transform , statistics , physics , quantum mechanics , random variable , philosophy , linguistics , geometry
The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to ‘identically zero’ expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x →0, allows us to compute the exponentially small term that these ‘identically zero’ expansions represent.