The Origin of the Method of Steepest Descent

The method of steepest descent, also known as the saddle-point method, is a natural development of Laplace's method applied to the asymptotic estimate of integrals of analytic functions. Mathematicians have often attributed the method of steepest descent to the physicist Peter Debye, who in 1909 worked it out in an asymptotic study of Bessel functions. Debye himself remarked that he had borrowed the idea of the method from an 1863 paper of Bernhard Riemann. The present article offers a detailed historical analysis of the creation of the method of steepest descent. We show that the method dates back to Cauchy and that, 25 years before Debye, the Russian mathematician Pavel Alexeevich Nekrasov had already used this technique and extended it to more general cases. Copyright 1997 Academic Press. La methode de la descente la plus rapide, que l'on appelle actuellement methode du col, represente un developpement naturel de la methode de Laplace pour l'estimation asymptotique des integrales des fonctions analytiques. La methode du col est generalement liee au nom du physicien Peter Debye, qui a utilise en 1909 les idees que Riemann a presentees dans son article de 1863 afin de donner une analyse asymptotique des fonctions de Bessel. Le present travail donne une etude bien detaillee de la creation de la methode du col. Tout d'abord nous montrons que la methode du col remonte aux travaux de Cauchy et ensuite que le mathematicien russe Pavel Alexeevich Nekrasov a applique cette methode un quart de siecle avant Debye et l'a etendue aux cas generaux. Copyright 1997 Academic Press. [FORMULA]