Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

stationary, the magnitude of any factor,g(z), multiplying the exponential function,

exp[f(z)]∼exp[Aeiα(z−z 0 )^2 ], is at least comparable to its magnitude elsewhere,

then this result can be used to obtain an approximation to the value of the

integral ofh(z)=g(z) exp[f(z) ]. This is the basis of the method of stationary

phase.

Returning to the behaviour of a function exp[f(z) ] at one of its saddle points,

we can now see how the considerations of the previous paragraphs can be applied

there. We already know, from equation (25.62) and the discussion immediately

following it, that in the equation

h(z)≈g(z 0 ) exp(f 0 )exp{^12 Aρ^2 [cos(2θ+α)+isin(2θ+α)]}

(25.71)

the second exponent is purely imaginary on a level line, and equal to zero at

the saddle point itself. What is more, since∇ψ= 0 at the saddle, the phase is

stationary there; on one level line it is a maximum and on the other it is a

minimum. As there are two level lines through a saddle point, a path on which

the amplitude of the integrand is constant could go straight on at the saddle

point or it could turn through a right angle. For the moment we assume that it

runs continuously through the saddle.

On the level line for which the phase at the saddle point is a minimum, we can

write the phase ofh(z) as approximately

argg(z 0 )+Imf 0 +v^2 ,

wherevis real,iv^2 =^12 Aeiα(z−z 0 )^2 and, as previously,Aeiα=f′′(z 0 ). Then

eiπ/^4 dv=±

√

A

2

eiα/^2 dz, (25.72)

leading to an approximation to the integral of

∫

h(z)dz≈±g(z 0 ) exp(f 0 )

∫∞

−∞

exp(iv^2 )

√

A

2

exp[i(^14 π−^12 α)]dv

=±g(z 0 ) exp(f 0 )

√

πexp(iπ/4)

√

A

2

exp[i(^14 π−^12 α)]

=±

√

2 π

A

g(z 0 ) exp(f 0 ) exp[^12 i(π−α)]. (25.73)

Result (25.68) was used to obtain the second line above. The±ambiguity is again

resolved by the directionθof the contour; it is positive if− 3 π/ 4 <θ≤π/4;

otherwise, it is negative.

What we have ignored in obtaining result (25.73) is that we have integrated

along a level line and that therefore the integrand has the same magnitude far

from the saddle as it has at the saddle itself. This could be dismissed by referring

to the fact that contributions to the integral from the ends of the Cornu spiral