Evaluating $P\int_{-\infty}^{\infty}\exp(imx^2}dx$

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Homework Statement


Evaluate $P\int_{-\infty}^{\infty}\exp(imx^2}dx$, $m>0$.

Homework Equations


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The Attempt at a Solution


Consider the following contour $C$ consisting of
$C_1:=\{z=x+iy:y=0, x:-s \to s\}$
$C_2:=\{z=x+iy:x=0, y:0 \to is\}$
$C_3:=\{z=x+iy:x: s \to -s, y: is \to -is \}$
$C_4:=\{z=x+iy:x=0: y:-is \to 0\}$

This is where i am stuck
 
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Sorry, i gave the wrong title
 
Well, why was that contour chosen? Does the integral go to 0 on some segments?
 
actually in C_2, x=s and in C_4, x=-s
 
Ok, I worked so far as that it only remains to show that the integrals on C_2 and C_4 are zero. I suspect having to use an ML-bound but don't really know how to do this.
 
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