Method of steepest descent

In mathematics, the method of steepest descent or saddle-point method is an extension of <a href="/facts/Laplace%2527s_method/Hr6YjG6o">Laplace's method</a> for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (<a href="/facts/Saddle_point/v5MT6YbM">saddle point</a>), 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 the following conditions hold:

<ol><li>C′ passes through one or more zeros of the derivative g′(z),</li>
<li>the imaginary part of g(z) is constant on C′.</li></ol>
The method of steepest descent was first published by Debye (1909), who used it to estimate <a href="/facts/Bessel_functions/hSyvFPqC">Bessel functions</a> and pointed out that it occurred in the unpublished note by Riemann (1863) about <a href="/facts/Hypergeometric_functions/yTEC6NBJ">hypergeometric functions</a>. The contour of steepest descent has a minimax property, see Fedoryuk (2001). Siegel (1932) described some other unpublished notes of Riemann, where he used this method to derive the <a href="/facts/Riemann%25E2%2580%2593Siegel_formula/sI6bzDRT">Riemann–Siegel formula</a>.

Method of steepest descent open-in-new

Method of steepest descent