Asylptotic Evaluation of Integrals

Abstract

Radiation and diffraction problems in open regions (with infinite cross section) are usually solved in terms of integral representations for the fields which cannot be evaluated in closed form. In many applications, however, the integrands contain a large parameter, to be called Q, in terns of which one may obtain an approximation to the integrals. While such an evaluation can be treated for rather general functional dependences of the integrand on Q, it will suffice within the present context to consider integrals of the following type:

$$ {\text{I}}\left( {\rm{\Omega }} \right) = \int\limits_{{\rm{\bar P}}_{\text{z}} } {{\text{f}}\left( {\text{z}} \right){\text{e}}^{{\rm{\Omega q}}\left( {\text{z}} \right)} \,{\text{dz}}} \,\,\,, $$

((1))

where f and q are analytic functions of the complex variable z along the path of integration ${{\rm{\bar P}}_{\text{z}} }$, whose endpoints lie at infinity, and where the large parameter Q is assumed to be positive.