The Origin of the Method of Steepest Descent

✍ Scribed by Svetlana S. Petrova; Alexander D. Solov'ev

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 301 KB
• Volume: 24
• Category: Article
• ISSN: 0315-0860
• DOI: 10.1006/hmat.1996.2146

✦ Synopsis

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.