Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.8 APPROXIMATIONS TO INTEGRALS

any of the continuations to infinity of the level lines that pass through the saddle.

In practical applications the start- and end-points of the path are nearly always

at singularities off(z) with Ref(z)→−∞and|h(z)|→0.

We now set out the complete procedure for the simplest form of integral

evaluation that uses a method of steepest descents. Extensions, such as including

higher terms in the Taylor expansion or having to pass through more than one

saddle point in order to have appropriate termination points for the contour, can

be incorporated, but the resulting calculations tend to be long and complicated,

and we do not have space to pursue them in a book such as this one.

As our general integrand we take a function of the formg(z)h(z), where, as

before,h(z)=exp[f(z) ]. The functiong(z) should neither vary rapidly nor have

zeros or singularities close to any saddle point used to evaluate the integral.

Rapidly varying factors should be incorporated in the exponent, usually in the

form of a logarithm. Providedg(z) satisfies these criteria, it is sufficient to treat

it as a constant multiplier when integrating, assigning to it its value at the saddle

point,g(z 0 ).

Incorporating this and retaining only the first two non-vanishing terms in

equation (25.61) gives the integrand as

g(z 0 ) exp(f 0 ) exp[^12 Aeiα(z−z 0 )^2 ]. (25.64)

From the way in which it was defined, it follows that on the l.s.d. the imaginary

part off(z)isconstant(=Imf 0 ) and that the final exponent in (25.64) is either

zero (atz 0 ) or negative. We can therefore write it as−s^2 ,wheresis real. Further,

since exp[f(z)]→0 at the start- and end-points of the contour, we must have

thatsruns from−∞to +∞, the sense ofsbeing chosen so that it is negative

approaching the saddle and positive when leaving it.

Making this change of variable,

1

2 Ae

iα(z−z

0 )

(^2) =−s (^2) ,withdz=±
√
2
A
exp[^12 i(π−α)]ds,
(25.65)
allows us to express the contribution to the integral from the neighbourhood of
the saddle point as
±g(z 0 ) exp(f 0 )
√
2
A
exp[^12 i(π−α)]
∫∞
−∞
exp(−s^2 )ds.
The simple saddle-point approximation assumes that this is the only contribution,
and gives as the value of the contour integral
∫
C
g(z) exp[f(z)]dz=±
√
2 π
A
g(z 0 ) exp(f 0 ) exp[^12 i(π−α)],
(25.66)
where we have used the standard result that
∫∞
−∞exp(−s
(^2) )ds=√π. The overall±