Steepest Descents, Method of

(or saddle-point method), a method for finding asymptotic expressions for certain integrals. Many special functions, such as cylindrical functions and spherical functions, can be expressed as integrals of the form

ʃ_ce^zf(τ)dτ

where f(τ) = u(x, y) + iv(x, y) is an analytic function of τ = ξ + iy such that u(x, y) approaches — ∞ at either end of the contour C. In order to compute these integrals for large positive values of z, the method of steepest descents is used.

The method consists in deforming the contour C into a contour C” that has the same endpoints as C and passes through a zero τ₀ of the function f’ (τ) along a curve of the form v(x, y) = constant; according to the Cauchy integral theorem, the value of the integral does not change under the deformation of the contour. On the surface t = u(x, y) the contour C is represented by a path that passes through a saddle point of the surface in such a way that on both sides of this point the path descends as steeply as possible (hence the name of the method) toward large negative values of u(x, y). Therefore, when z is real and positive, only the immediate neighborhood of the point τ₀ has any substantial effect on the value of the integral (*). This circumstance can be made use of to obtain asymptotic expressions for the integral—for example, by replacing the function f(τ) by a portion of its Taylor series in the neighborhood of the point τ₀.

The method of steepest descents as a rule permits a complete asymptotic expansion to be found for the integral (*).

If the integrand is a multiple-valued function, then in the deformation of the contour it is necessary to consider the cuts arising as a result of the multivaluedness and to direct part of the path along the cuts. The method of steepest descents can also be applied to the computation of integrals of the form

ʃ_c (τ – τ)^{α – 1}F (τ)e^zf(τ)dτ