Using Cauchy's Theorem to Solve the Complex Integral of cos(ax^2)

[metgt4]

Homework Statement

By applying Cauchy's theorem to a suitable contour, prove that the integral of cos(ax²) = (pi/8a)^1/2

Homework Equations

Cauchy's integral formula:

http://en.wikipedia.org/wiki/Cauchy's_integral_formula

The Attempt at a Solution

I'm not sure where to go after what I've got scanned. I've tried a few other things, but I can't seem to get anywhere that looks like something I could solve.

 

[Count Iblis]

Hint: Consider the function exp(-a z^2).

 

[metgt4]

What I do know about e^{iaz^2} is that within 0 < argz < pi/4 it is analytic and
e^{iaz^2}-->0 as |z|-->infinity

Could I expand that in terms of a Taylor series and use the method of residues to solve the integral? Would this be the correct expansion for cos(ax²):

e^x = 1 + x + x²/2! + x³/3!

Taylor series for x = iaz² added to the taylor series for x = -iaz²
= 1/2(e^{iaz^2} + e^{-iaz^2}) = 1/2[(1 + iaz² - a²z⁴/2 - a³z⁶/6 +...) + (1 - iaz + a²z⁴/2 + a³z⁶/6 +...) = 1/2(2) = 1?

and then would I express 1 as a complex exponential?

Is that right?