Integral of |f(k)|^2: Proving Equality to 1

[Wishbone]

ok so here's the question, show explicitly that

the integral from -inf to inf of |f(k)|^2=1

where f(x) = \frac{N}{\sqrt{\sigma}}*e^{\frac{-x^2}{2\sigma^2}}
When doing the integral for the forier transform, I was going to use the gaussian integral to simplify it, but I don't htink I can do that, any ideas?

 

[Wishbone]

I have tried and looked up several methods to solving this awfully ugly integral, anyone have any ideas?

 

[J77]

Only a suggesstion - but it looks like you're going to have a situation where:

limit as {k\rightarrow\pm \infty} of \frac1{k}e^{-k^2} with some constants thrown in...

my suspicion is that you'll have to use L'Hopital.

 

[Hurkyl]

It is a Gaussian integral...

 

[Wishbone]

Im not sure how it is, I can see that f(x) might be, but not f(K)

 

[Hurkyl]

What's the difference between f(x) and f(k)?

 

[Wishbone]

a factor of e^-(pi)*i*k*x

 

[Wishbone]

anyone?...

 

[HallsofIvy]

Once again, what's the difference between f(x) and f(k)? f(k) is just f(x) with variable k instead of x!

 

[Wishbone]

I don't see that, I don't understand how the Fourier transform just changes the x's to k's.

 

[nrqed]

ok so here's the question, show explicitly that

the integral from -inf to inf of |f(k)|^2=1

where f(x) = \frac{N}{\sqrt{\sigma}}*e^{\frac{-x^2}{2\sigma^2}}
When doing the integral for the forier transform, I was going to use the gaussian integral to simplify it, but I don't htink I can do that, any ideas?

This is a bit confusing. Do you really mean f(k)?? (in which case it is the function you gave with x replaced by k) or do you mean F(k), the Fourier transform of f(x)? (I am assuming that you mean the latter otherwise the question has nothing to do with Fourier transforms and the question is trivial. I think this is what you meant and that the other posters missed).

well, you have to calculate F(k), the Fourier transform of your f(x) first. Do you know how to calculate a Fourier transform in the first place? If not, you should look up the definition and then ask more questions if thsi is not clear.

 

[Wishbone]

I am given f(k). Nothing else is made clear to me. I do infact know how to do Fourier transforms, I do infact could solve the problem if it, or the help i was given hear made any sense.

 

[shmoe]

I have tried and looked up several methods to solving this awfully ugly integral, anyone have any ideas?

Can you latex what this integral is? I'm confused on where your confusion lies, so knowing what you are staring at would help us out.