# Homework Help: Complex Integral Problem

1. Feb 8, 2010

### metgt4

1. The problem statement, all variables and given/known data

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

2. Relevant equations

Cauchy's integral formula:

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

3. 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.

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2. Feb 8, 2010

### Count Iblis

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

3. Feb 9, 2010

### metgt4

What I do know about eiaz^2 is that within 0 < argz < pi/4 it is analytic and
eiaz^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):

ex = 1 + x + x2/2! + x3/3!

Taylor series for x = iaz2 added to the taylor series for x = -iaz2
= 1/2(eiaz^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?