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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 saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form : where ''C'' is a contour and λ is large. One version of the method of steepest descent deforms the contour of integration so that it passes through a zero of the derivative ''g′''(''z'') in such a way that on the contour ''g'' is (approximately) real and has a maximum at the zero. The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note about hypergeometric functions. The contour of steepest descent has a minimax property, see . described some other unpublished notes of Riemann, where he used this method to derive the Riemann-Siegel formula. ==A simple estimate〔A modified version of Lemma 2.1.1 on page 56 in .〕== Let and . If : where denotes the real part, and there exists a positive real number such that : then the following estimate holds: : Proof of the simple estimate : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Method of steepest descent」の詳細全文を読む スポンサード リンク
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