Why Is My Fourier Transform of a Gaussian Incorrect?

[jaurandt]

Attached is a personal problem that I spent last night working on for about 2 hours and something is going wrong, I just can not figure it out what. The answer by the big X is what I wound up with but it's obviously not correct. Could someone please guide me through solving this? The starting problem is on the top left, the correct answer is boxed on the top right.

 

[PeterDonis]

Please use the PF LaTeX feature to post your equations directly in the thread. Per PF rules, equations should not be posted as images, as that makes it impossible to quote them in replies.

Help on the LaTeX feature can be found here:

https://www.physicsforums.com/help/latexhelp/

 

[jaurandt]

Please use the PF LaTeX feature to post your equations directly in the thread. Per PF rules, equations should not be posted as images, as that makes it impossible to quote them in replies.

Help on the LaTeX feature can be found here:

https://www.physicsforums.com/help/latexhelp/

Sorry, I will do so from now on. But what's most important to me is the problem for the sake of my sanity...

 

[Cryo]

This is a Fourier transform of a Gaussian. Plenty of online resources for it:

https://duckduckgo.com/?q=fourier+transform+of+a+gaussian&t=ffab&ia=web
Simply complete the square under the integral and change the coordinates to get to the standard Gaussian integral. I am not going to repeat it here as there is plenty of explanations online, e.g.

http://www.thefouriertransform.com/pairs/gaussian.php

 

[jaurandt]

This is a Fourier transform of a Gaussian. Plenty of online resources for it:

https://duckduckgo.com/?q=fourier+transform+of+a+gaussian&t=ffab&ia=web
Simply complete the square under the integral and change the coordinates to get to the standard Gaussian integral. I am not going to repeat it here as there is plenty of explanations online, e.g.

http://www.thefouriertransform.com/pairs/gaussian.php

Thank you!