Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

angles) in which the phase ofh(z) is independent ofρ. They make angles ofπ/ 4

with the level lines throughz 0 and are given by

θ=−^12 α, θ=^12 (±π−α),θ=π−^12 α.

From our previous discussion it follows that these four directions will be the

lines of steepest descent (or ascent) on moving away from the saddle point. In

particular, the two directions for which the term cos(2θ+α) in (25.62) isnegative

will be the directions in which|h(z)|decreasesmost rapidly from its value at the

saddle point. These two directions are antiparallel, and a steepest descents path

following them is a smooth locally straight line passing the saddle point. It is

known as theline of steepest descents(l.s.d.) through the saddle point. Note that

‘descents’ is plural as on this line the value of|h(z)|decreases onbothsides of the

saddle. This is the line which we will make the path of the contour integral of

h(z) follow. Part of a typical l.s.d. is indicated by the dashed line in figure 25.12.

25.8.2 Steepest descents method

To help understand how an integral along the line of steepest descents can be

handled in a mechanical way, it is instructive to consider the case where the

functionf(z)=−βz^2 andh(z)=exp(−βz^2 ). The saddle point is situated at

z=z 0 = 0, withf 0 =f(z 0 )=1andf′′(z 0 )=− 2 β, implying thatA=2|β|and

α=±π+argβ,withthe±sign chosen to putαin the range 0≤α< 2 π.Then

the l.s.d. is determined by the requirement that sin(2θ+α) = 0 whilst cos(2θ+α)is

negative; together these imply that, for the l.s.d.,θ=−^12 argβorθ=π−^12 argβ.

Since the Taylor series forf(z)=−βz^2 terminates after three terms, expansion

(25.61) for this particular function is not an approximation toh(z), but is exact.

Consequently, a contour integral starting and endingin regions of the complex

plane where the function tends to zero and following the l.s.d. through the saddle

point atz= 0 will not only have a straight-line path, but will yield an exact

result. Settingz=te−

1

2 argβwill reduce the integral to that of a Gaussian function:

e−

(^12) argβ
∫∞
−∞
e−|β|t
2
dt=e−
(^12) argβ
√
π
|β|
.
The saddle-point method for a more general function aims to simulate this
approach by deforming the integration contourCand forcing it to pass through
a saddle pointz=z 0 , where, whatever the function, the leadingz-dependent term
in the exponent will be a quadratic function ofz−z 0 , thus turning the integrand
into one that can be approximated by a Gaussian.
The path well away from the saddle point may be changed in any convenient
way so long as it remains within the relevant sectors, as determined by the end-
points ofC. By a ‘sector’ we mean a region of the complex plane, any part of
which can be reached from any other part of the same region without crossing