A Generalization of the Method of Steepest Descent

Résumé du livre

The authors consider the asymptotic expansion as lambda approaches infinity of contour integrals with integrand H(lambda w(z)) g(z), H(t) being an entire function of t. When H is the exponential function, the asymptotic expansion of such an integral can be obtained by the method of steepest descent. The generalization of that method has two major new features. Firstly, the zeros of w(z) are critical points of the integrand. Secondly, there exist points from which no path of descent can be drawn; i.e., there exist curves which are boundaries between two hills of H(lambda w(z)). These features lead to new types of expansions for which the Mellin transform technique of Handelsman and Lew is ideally suited. The authors develop the general method and treat specific examples in detail. (Author).