# Contour integral involving gamma function

1. Sep 30, 2014

### CAF123

1. The problem statement, all variables and given/known data
Evaluate the integral by closing a contour in the complex plane $$\int_{-\infty}^{\infty} dx e^{iax^2/2}$$

2. Relevant equations
Residue theorem

3. The attempt at a solution
My initial choice of contour was a semicircle of radius R and a line segment from -R to R. In the limit R to infinity, I would hopefully recover the integral in OP. But then I realized this was going to work because the integral over the semicircle would not vanish when R tended to infinity. My question is, what is the best way to see what the best contour should be for a given problem and how to determine the one right for this problem?

Many thanks

2. Sep 30, 2014

### haruspex

By symmetry, you could conside only a semi infinite integral.
Because of the exponential term, I suggest a rectangle will work better than an arc.

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