Method of steepest descent

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In mathematics, the method of steepest descent 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 ∫ C f ( z ) e λ g ( z ) d z , {\displaystyle \int _{C}f(z)e^{\lambda g(z)}\,dz,} where C is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that...

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