A general formula for the method of steepest descent using Perron’s formula

where f and g are (usually) analytic functions, k is (usually) a real parameter. The general idea is to use Cauchy’s theorem which allows deformation of contours in the complex plane to simplify the problem. The idea was proposed by Debye, and the basic idea is to choose a new contour D such that • D passes through one of more zeros of f ′(z) • The imaginary part of f(z) is constant on D Since the imaginary part of the integral is constant, then if f(z) = g(z) + ih then we instead need to evaluate. I(k)e ∫