Method of steepest descent

Summary

The method of steepest descent, also called the saddle point method, finds the asymptotic behaviour of contour integrals $\int_C g(z)\, e^{\lambda f(z)}\,dz$ as the parameter $\lambda$ grows. One deforms the contour so that it passes through saddle points of the holomorphic function $f$ along paths on which $\operatorname{Im} f$ is constant. On these paths the integrand does not oscillate, and Laplace's method gives the leading terms. It yields the asymptotics of many special functions, such as the Airy and Bessel functions, and of coefficients of generating functions.