Method of steepest descent

For the optimization algorithm, see Gradient descent.

In mathematics, the method of steepest descent or stationary phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase.

The method of steepest descent has been rediscovered several times. Riemann used it in about 1859 to derive the Riemann-Siegel formula, and it was rediscovered by Debye (1909).

References

• Debye, P (1909), "Näherungsformeln für die Zylinderfunktionen für große Werte des Arguments und unbeschränkt veränderliche Werte des Index", Mathematische Annalen, 67 (4): 535–558, doi:10.1007/BF01450097
• Deift, P.; Zhou, X. (1993), "A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation", Ann. of Math., vol. 137, no. 2, pp. 295–368, doi:10.2307/2946540.
• Erdelyi, A. (1956), Asymptotic Expansions, Dover.
• Fedoryuk, M.V. (2001) [1994], "Method of steepest descent", Encyclopedia of Mathematics, EMS Press