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# Fourier transform watch

1. So I'm having some trouble getting on top of fourier transforms.

and

since we have an even function.

But... I feel like I'm missing something major... the integral I'm required to do seems massive... is there an easier way that I'm missing?
2. It's not clear from what you've written what you know, what it is you are supposed to show, etc.

For this particular case, there's a trick (which it would be very hard to come up with yourself);

Differentiate f(x); you'll get something like f'(x) = Ax f(x) for some constant A (based on gamma).
Now, take the FT of both sides, recalling that there are rules for taking the FT of a derivative, and for finding the FT of x f(x) in terms of the derivative of the FT of f.

This gives you a differential equation that your Fourier transform must follow; solve it and you're done. (For the arbit constant, note that you know the value of the FT when s = 0).
3. Got it, thanks!
4. Ah, yes - well, I used a different method (it seems it was in my notes; it's the Gaussian function) and I get to the last stage (where I have to use the integral that I'm given) and I can't seem to understand the last step (ie, substitution... lol)

(Here I completed the square and took out the exponential that's only in terms of and ).

Now, if I let and use , then I should get the correct answer, which is

However, the problem I've got is that my doesn't cancel and I need a on the top of the fraction, too. Have I gone wrong somewhere in my calculation? I can't spot it.
5. When you complete the square, you are left with , not .

As far as the sqrt(2), the 1/2 in should provide it.

The reason I wasn't too keen to advocate the solution: you are tacitly assuming that , which I think will be hard without contour integration.
6. (Original post by DFranklin)
When you complete the square, you are left with , not .

But isn't ? The x coefficient needs to be , from what I've written above, anyway.
7. Yes, sorry. (Doesn't seem to be my best day for silly mistakes).

You need , and so

, not
8. Yes, sorry. (Doesn't seem to be my best day for silly mistakes).

You need , and so

, not
9. (Original post by DFranklin)
Yes, sorry. (Doesn't seem to be my best day for silly mistakes).

You need , and so

, not
But we've got

Right?
10. I'm not sure what your question is. In your original post, you said you wanted to make the substitution . I was pointing out why this was going to be wrong.
11. Nevermind, finally realised what I was doing. I'm not a late night person. Thanks very much, helped a lot!

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