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 the following conditions hold:
C′ passes through one or more zeros of the derivative g′(z), the imaginary part of g(z) is constant on C′. The method of steepest descent was first published by Debye (1909), who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note by Riemann (1863) about hypergeometric functions. 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 Siegel (1932) described some other unpublished notes of Riemann, where he used this method to derive the Riemann–Siegel formula. When applied to the distribution of the sum of a large number of independent random variables in statistics, the method of steepest descent is called the saddle-point approximation method. In statistical physics, this approximation is widely used in the thermodynamic limit to evaluate partition functions and free energies.
Basic idea The method of steepest descent is a method to approximate a complex integral of the form I ( λ ) = ∫ C f ( z ) e λ g ( z ) d z {\displaystyle I(\lambda )=\int _{C}f(z)e^{\lambda g(z)}\,\mathrm {d} z} for large λ → ∞ {\displaystyle \lambda \rightarrow \infty } , where f ( z ) {\displaystyle f(z)} and g ( z ) {\displaystyle g(z)} are analytic functions of z {\displaystyle z} . Because the integrand is analytic, the contour C {\displaystyle C} can be deformed into a new contour C ′ {\displaystyle C'} without changing the integral. In particular, one seeks a new contour on which the imaginary part, denoted ℑ ( ⋅ ) {\displaystyle \Im (\cdot )} , of g ( z ) = ℜ [ g ( z ) ] + i ℑ [ g ( z ) ] {\displaystyle g(z)=\Re [g(z)]+i\,\Im [g(z)]} is constant ( ℜ ( ⋅ ) {\displaystyle \Re (\cdot )} denotes the real part). Then I ( λ ) = e i λ ℑ { g ( z ) } ∫ C ′ f ( z ) e λ ℜ { g ( z ) } d z , {\displaystyle I(\lambda )=e^{i\lambda \Im \{g(z)\}}\int _{C'}f(z)e^{\lambda \Re \{g(z)\}}\,\mathrm {d} z,} and the remaining integral can be approximated with other methods like Laplace's method.
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