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Explicit Series Formulae for the Evaluation of Integrals by the Method of Steepest Descents
Author(s) -
Campbell John A.,
Fröman Per Olof,
Walles Erik
Publication year - 1987
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 - 10.1002/sapm1987772151
Subject(s) - saddle point , mathematics , series (stratigraphy) , saddle , zero (linguistics) , order (exchange) , point (geometry) , explicit formulae , mathematical analysis , pure mathematics , mathematical optimization , geometry , paleontology , linguistics , philosophy , finance , economics , biology
Explicit series formulae are given for use in the method of steepest descents, where it is required to find the contribution to an integral of the form ∫e vw ( z ) φ ( z ) dz from a well‐isolated saddle point z 0 . Here v is a large positive parameter, and w ( N ) ( z 0) ≠ 0 while lower‐order derivatives at z 0 are equal to zero. Examples where the formulae are helpful are presented.