Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

Finally, putting the various values into the formula yields

F(x)∼+

(

2 π

A

) 1 / 2

g(i)exp[f(i)]exp[^12 i(π−α)]

=+

(

2 π

2 x^3 /^2

) 1 / 2

x^1 /^2

2 π

exp

(

−

2

3

x^3 /^2

)

exp[^12 i(π−π)]

=

1

2

√

πx^1 /^4

exp

(

−

2

3

x^3 /^2

)

.

This is the leading term in the asymptotic expansion ofF(x),which,asshowninequation
(25.39), is a particular contour integral solution of Stokes’ equation. The fact that it tends
to zero in a monotonic way asx→+∞allows it to be identified with the Airy function,
Ai(x).
We may ask why the saddle point att=−iwas not used. The answer to this is as
follows. Of course, any path that starts and ends in the right sectors will suffice, but
if another saddle point exists close to the one used, then the Taylor expansion actually
employed is likely to be less effective than if there were no other saddle points or if there
were only distant ones.
An investigation of the same form as that used att=+ishows that the saddle att=−i
is higher by a factor of exp(^43 x^3 /^2 ) and that its l.s.d. is orientated parallel to the imaginary
t-axis. Thus a path that went through it would need to go via a region of largish negative
imaginaryt, over the saddle att=−i, and then, when it reached the col att=+i,bend
sharply and follow part of the same l.s.d. as considered earlier. Thus the contribution from
thet=−isaddle would be incomplete and roughly half of that from thet=+isaddle
would still have to be included. The more serious error would come from the first of these,
as, clearly, the part of the path that lies in the plane Ret= 0 is not symmetric and is far
from Guassian-like on the side nearer the origin. The Gaussian-path approximation used
will therefore not be a good one, and, what is more, the resulting error will be magnified by
a factor exp(^43 x^3 /^2 ) compared with the best estimate. So, both on the grounds of simplicity
and because the effect of the other (neglected) saddle point is likely to be less severe, we
choose to use the one att=+i.

25.8.3 Stationary phase method

In the previous subsection we showed how to use the saddle points of an

exponential function of a complex variable to evaluate approximately a contour

integral of that function. This was done by following the lines of steepest descent

that passed through the saddle point; these are lines on which the phase of the

exponential is constant but its amplitude is varying at the maximum possible

rate for that function. We now introduce an alternative method, one that entirely

reverses the roles of amplitude and phase. To see how such an alternative approach

might work, it is useful to study how the integral of an exponential function of

a complex variablecan be represented as the sum of infinitesimal vectors in the

complex plane.

We start by studying the familiar integral

I 0 =

∫∞

−∞

exp(−z^2 )dz, (25.67)