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Generalization of Watson's Lemma

Published online by Cambridge University Press:  20 November 2018

R. Wong  and
M. Wyman
R. Wong
Affiliation:
The University of Alberta, Edmonton, Alberta
M. Wyman
Affiliation:
The University of Alberta, Edmonton, Alberta
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Many functions, F(z), have integral representations of the form

1.1

the so-called Laplace transform of f(t). When f(t) satisfies certain regularity conditions, it is possible to use Cauchy's theorem to deform the contour so that F(z) has the integral representation

1.2

where 7 is a fixed real number, and the path of integration is the straight line joining t = 0 to t = ∞ etr, small indentations in the path of integration being allowed to avoid singularities of f(t) where necessary. When the two functions defined by (1.1) and (1.2) are not the same, (1.2) provides the more general situation for a theoretical discussion of the properties of F(z).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 185 - 208
Copyright
Copyright © Canadian Mathematical Society 1972

References

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