Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.8 APPROXIMATIONS TO INTEGRALS

From the observations contained in the two previous paragraphs, we deduce

that a path that follows the lines of steepest descent (or ascent) can never form

a closed loop. On such a path,φ, and hence|h(z)|, must continue to decrease

(increase) until the path meets a singularity off(z). It also follows that if a level

line ofh(z) forms a closed loop in the complex plane, then the loop must enclose

a singularity off(z). This may (ifφ→∞) or may not (ifφ→−∞) produce a

singularity inh(z).

We now turn to the study of the behaviour ofh(z) at a saddle point and how

this enables us to find an approximation to the integral ofh(z)alongacontour

that can be deformed to pass through the saddle point. At a saddle pointz 0 ,at

whichf′(z 0 ) = 0, both∇φand∇ψare zero, and consequently the magnitude and

phase ofh(z) are both stationary. The Taylor expansion off(z) at such a point

takes the form

f(z)=f(z 0 )+0+

1

2!

f′′(z 0 )(z−z 0 )^2 +O(z−z 0 )^3. (25.60)

We assume thatf′′(z 0 )= 0 and write it explicitly asf′′(z 0 )≡Aeiα, thus defining

the real quantitiesAandα.Ifithappensthatf′′(z 0 ) = 0, then two or more

saddle points coalesce and the Taylor expansion must be continued until the first

non-vanishing term is reached; we will not consider this case further, though the

general method of proceeding will be apparent from what follows. If we also

abbreviate the (in general) complex quantityf(z 0 )tof 0 , then (25.60) takes the

form

f(z)=f 0 +^12 Aeiα(z−z 0 )^2 +O(z−z 0 )^3. (25.61)

To study the implications of this approximation forh(z), we writez−z 0 as

ρeiθwithρandθboth real. Then

|h(z)|=|exp(f 0 )|exp[^12 Aρ^2 cos(2θ+α)+O(ρ^3 )]. (25.62)

This shows that there are four values ofθfor which|h(z)|is independent ofρ(to

second order). These therefore correspond to two crossing level lines given by

θ=^12

(

±^12 π−α

)

andθ=^12

(

±^32 π−α

)

. (25.63)

The two level lines cross at right angles to each other. It should be noted that

the continuations of the two level lines away from the saddle are not straight

in general. At the saddle they have to satisfy (25.63), but away from it the lines

must take whatever directions are needed to make∇φ= 0. In figure 25.12 one of

the level lines (|h|= 1) has a continuation (y= 0) that is straight; the other does

not and bends away from its initial directionx=1.

So far as the phase ofh(z) is concerned, we have

arg[h(z)] = arg(f 0 )+^12 Aρ^2 sin(2θ+α)+O(ρ^3 ),

which shows that there are four other directions (two lines crossing at right