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Problem
F denotes a forward fourier transform, the variables I'm transforming between are x and k
 See attachment
Relevant equations
So first of all I note I am given a result for a forward fourier transform and need to use it for the inverse one.
The result I am given to use, written out is :
## \int e^{ikx} e^{\alpha x^{2}} dx = \frac{1}{2\alpha} e^{k^{2}/4\alpha} ##
I note that ## F(k) C(t) ## gives me a function of k, so I apply ## F^{1} ## on this I get a function of x.
Attempt:
My thoughts are I'm looking to change signs in my exponential terms so that it is , effectively a forward transform and then use the result, so just thinking of it as a integration result rather than a particular fourier transform.
However if I do this my exponential terms are:
## e^{(ikx+\alpha k^{2})} ##
, I don't know how I can then apply the result, completing the square looks like the only candidate to me , but this seems like it will be too scrappy, in particular with 'i' terms, how do I deal with this ## e^{ikx} ## term, if I'm on the right lines, is completion of the square necessary or is there some other approach to being able to use this result ?
Many thanks in advance.
F denotes a forward fourier transform, the variables I'm transforming between are x and k
 See attachment
Relevant equations
So first of all I note I am given a result for a forward fourier transform and need to use it for the inverse one.
The result I am given to use, written out is :
## \int e^{ikx} e^{\alpha x^{2}} dx = \frac{1}{2\alpha} e^{k^{2}/4\alpha} ##
I note that ## F(k) C(t) ## gives me a function of k, so I apply ## F^{1} ## on this I get a function of x.
Attempt:
My thoughts are I'm looking to change signs in my exponential terms so that it is , effectively a forward transform and then use the result, so just thinking of it as a integration result rather than a particular fourier transform.
However if I do this my exponential terms are:
## e^{(ikx+\alpha k^{2})} ##
, I don't know how I can then apply the result, completing the square looks like the only candidate to me , but this seems like it will be too scrappy, in particular with 'i' terms, how do I deal with this ## e^{ikx} ## term, if I'm on the right lines, is completion of the square necessary or is there some other approach to being able to use this result ?
Many thanks in advance.
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