Fourier Transform of Gaussian Function

  • #1
402
1
Hello guys,

I have got a homework for Advanced Quantum Mechanics, actually I've tried to solve it in many ways my own, but I'm always forced to use computer at the end (For infinite series or improper integrals), I want to solve it my self, so I can do it also in the class!

The problem is:


Find the fourier transform of exp((a x^2)/2)

Actually the first way I tried is by splitting the fourier factor exp(ikx) to sine and cosine, by Euler form, but no use! because I reached "Abramowitz and Stegun integral", which have a result without details.

The second way is in the PDF, where I tried to release the exp((a x^2)/2) term to an infinite series, But also no use, I reached the Erf error function due to an infinite series!!!!

Thanks for your help :)
 

Answers and Replies

  • #2
olgranpappy
Homework Helper
1,271
3
Hello guys,

I have got a homework for Advanced Quantum Mechanics, actually I've tried to solve it in many ways my own, but I'm always forced to use computer at the end (For infinite series or improper integrals), I want to solve it my self, so I can do it also in the class!

The problem is:


Find the fourier transform of exp((a x^2)/2)

Actually the first way I tried is by splitting the fourier factor exp(ikx) to sine and cosine, by Euler form, but no use! because I reached "Abramowitz and Stegun integral", which have a result without details.

The second way is in the PDF, where I tried to release the exp((a x^2)/2) term to an infinite series, But also no use, I reached the Erf error function due to an infinite series!!!!

Thanks for your help :)

Try completeing the square in the exponent...

;)
 
  • #3
402
1
You mean in the PDF? or from the begining?
 
  • #4
402
1
Sorry, I didn't understand how can this benefit me, If I completed the square this will cause the appearance of a new exponential multiplicand!!!

I also failed because in the first term inside the exponential I have a, and in the second no a! so how can I do it?
 
  • #5
402
1
Why you guys aren't helping? Is the question too difficult? or I don't deserve the help??
 
  • #6
olgranpappy
Homework Helper
1,271
3
Why you guys aren't helping? Is the question too difficult? or I don't deserve the help??

calm down. I'm not online 24 hours a day. You completed the square in the exponent of the integrand of the fourier transform and what did you end up with. Show up your work, if possible. Also, if possible can you write your work down using LaTeX.
 
  • #7
402
1
Thank you :), It's solved, I succeeded, my mistake was in the completion of the square using variables, I used the constants and it worked ;)

Thanks :)
 

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