Some Extensions to the Method of Integration by Steepest Descents

Uniform Asymptotic Physical Optics Solutions for a Set of Diffraction Problems

“…To overcome this drawback it is convenient to regularise the integrand in (36). This procedure (to be referred to as the Multiplicative Method) was described in (Kouyoumjian & Pathak, 1974) as the modified Pauli-Clemmow method (Clemmow, 1950). It requires to introduce the regularised function…”

On a modification of Watson's Lemma

“…The solution of numerous problems in electromagnetic and acoustic wave propagation theory is frequently given by an integral r epresentation of the form (1) Here '1'( z) deno tes the desired property (for ins tance the electromagnetic or acoustic potential) . The integr ation in (1) is taken along a curve C in the complex v plane along which w (v) and u (v) are defin ed.…”

“…The integr ation in (1) is taken along a curve C in the complex v plane along which w (v) and u (v) are defin ed. These function contain generally cer tain parameters which depend on the physical configuration under inves tigation.…”

“…The parameter z is usually r ela ted to the di stance b etween ome point of r eference a nd the point of obse rvation. A reduction of the integral (1) to an asymptotic series has to be carried out when approximate express ions for large z are desired. The frequently used method of steepest descents [3, p. 86]1 b ecomes invalid w11en Ol1e of tlle above mentioned parameters ass umes a valu e for whi cll a pole of w(v) coincides with a saddle point of u (v).…”

“…The frequently used method of steepest descents [3, p. 86]1 b ecomes invalid w11en Ol1e of tlle above mentioned parameters ass umes a valu e for whi cll a pole of w(v) coincides with a saddle point of u (v). M ethods to cope with such cases have b een worked out and disc ussed [1], [8], [9]. The fonner two give expansions whi cll proceed in terms of hypergeometric functions besides a " leading term", while the la tter consists of an error integral as leading term and an asymptotic series with inverse powers of z .…”