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

[Sam1234]

Homework Statement

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

Homework Equations

None

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

 

[Sam1234]

Sorry, i gave the wrong title

 

[mutton]

Well, why was that contour chosen? Does the integral go to 0 on some segments?

 

[Sam1234]

actually in C_2, x=s and in C_4, x=-s

 

[Sam1234]

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.