How to Solve a Contour Integral with exp(-z^2) Over a Rectangle?

[NT123]

Homework Statement

Integrate exp(-z^2) over the rectangle with vertices at 0, R, R + ia, and ia.

Homework Equations

int(0, inf)(exp(-x^2)) = sqrt(pi/2)

The Attempt at a Solution

I really don't have much of an idea here - the function is analytic so has no residues... The part from 0 to R is just the real integral, but for the other 3 sides I'm not too sure on how to proceed.

 

[ideasrule]

Isn't the contour integral equal to 0 if there are no poles?

 

[NT123]

Isn't the contour integral equal to 0 if there are no poles?

This is what I would have thought, but I'm supposed to be using the integral of e^(-z^2) to evaluate the real integral int(0,inf)((e^(-x^2))*cos(2ax)), which is apparently equal to
sqrt(pi)*exp(-a^2)/2.

 

[Count Iblis]

See here:

https://www.physicsforums.com/showthread.php?t=368440

for some explanations.

 

[ideasrule]

Ah, that makes much more sense.

If we want to integrate from R+ia to ia, just integrate e^(-z^2)dz=e^-(x+ia)^2 dx from R to 0. Do the same for the other 3 sides. You won't get an analytic answer, but that's OK; just write out the entire contour integral first and you'll see where this is going.