Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.8 APPROXIMATIONS TO INTEGRALS

to the l.s.d., i.e. on the imaginaryt-axis, provides some reassurance. Whetherμis

positive or negative,

h(5 +iμ) = exp(50 + 10iμ− 25 − 10 iμ+μ^2 )=exp(25+μ^2 ).

This is greater thanh(5) for allμand increases as|μ|increases, showing that

the integration path really does lie at a minimum ofh(t) for a traversal in this

direction.

We now give a fully worked solution to a problem that could not be easily

tackled by elementary means.

Apply the saddle-point method to the function defined by

F(x)=

1

π

∫∞

0

cos(^13 s^3 +xs)ds

to show that its form for large positive realxis one that tends asymptotically to zero, hence

enablingF(x)to be identified with the Airy function,Ai(x).

We first express the integral as an exponential function and then make the change of
variables=x^1 /^2 tto bring it into the canonical form

∫

g(t)exp[f(t)]dtas follows:

F(x)=

1

π

∫∞

0

cos(^13 s^3 +xs)ds

=

1

2 π

∫∞

−∞

exp[i(^13 s^3 +xs)]ds

=

1

2 π

∫∞

−∞

x^1 /^2 exp[ix^3 /^2 (^13 t^3 +t)]dt.

We now seek to find an approximate expression for this contour integral by deforming its
path along the realt-axis into one passing over a saddle point of the integrand. Considered
as a function oft, the multiplying factorx^1 /^2 / 2 πis a constant, and any effects due to the
proximity of its zeros and singularities to any saddle point do not arise.
The saddle points are situated where

0=f′(t)=ix^3 /^2 (t^2 +1) ⇒ t=±i.

For reasons discussed later, we choose to use the saddle point att=t 0 =i. At this point,

f(i)=ix^3 /^2 (−^13 i+i)=−^23 x^3 /^2 andAeiα≡f′′(i)=ix^3 /^2 (2i)=− 2 x^3 /^2 ,

and soA=2x^3 /^2 andα=π.
Now, expandingf(t) aroundt=iby settingt=i+ρeiθ, we have

f(t)=f(i)+0+

1

2!

f′′(i)(t−i)^2 +O[(t−i)^3 ]

=−

2

3

x^3 /^2 +

1

2

2 x^3 /^2 eiπρ^2 e^2 iθ+O(ρ^3 ).

For the l.s.d. contour that crosses the saddle point we need the second term in this last
line to decrease asρincreases. This happens ifπ+2θ=±π,i.e.ifθ=0orθ=−π(or
+π); thus, the l.s.d. through the saddle is orientated parallel to the realt-axis. Given the
initial contour direction, the deformed contour should approach the saddle point from the
directionθ=−πand leave it along the lineθ=0.Since−π/ 2 < 0 ≤π/2, the overall sign
of the ‘omnibus’ approximation formula is determined as positive.