# Homework Help: An integral arising from the inverse Fourier transform

1. Nov 24, 2012

### fluidistic

1. The problem statement, all variables and given/known data
For a physics problem I must take the inverse Fourier transform of 2 functions.
Namely I must compute the integral $\frac{1}{\sqrt{2\pi}}\int_{-\infty} ^\infty [A\cos (ckt)+B\sin (ckt)]e^{ikx}dk$.

2. Relevant equations
i is the complex number. t is greater or equal to 0. In fact it could also be negative, there should be no problem.

3. The attempt at a solution
So I tried to tackle $\int _{-\infty} ^\infty \cos (ckt)e^{ikx}dk$ first but ran out of ideas.
Integration by parts does not look promising. Probably some substitution I guess but I don't see it. Any tip would be appreciated.

2. Nov 24, 2012

### vela

Staff Emeritus
Look into the integral representation of the Dirac delta function.

3. Nov 24, 2012

### fluidistic

Hi and thanks for the tip vela. I have it under the eyes (http://dlmf.nist.gov/1.17), but I don't see how this can help.
Edit: I rewrote $A\cos (ckt)+B\sin (ckt)$ as $Ae^{ickt}+Be^{-ickt}$ which makes the integral diverge.

Last edited: Nov 24, 2012
4. Nov 24, 2012

### vela

Staff Emeritus
The integrals don't converge in the normal sense, but you can use 1.17.12 to recognize the appearance of the delta function.

5. Nov 24, 2012

### vela

Staff Emeritus
If you don't find that approach satisfying, you can try throwing in a convergence factor and then taking a limit:
$$\lim_{\lambda \to 0^+} \int_{-\infty}^\infty \cos (ckt) e^{ikx} e^{-\lambda |k|}\,dk$$ You have to be a bit careful when taking the limit so that you end up with the delta functions.

6. Nov 24, 2012

### fluidistic

Thanks a lot! It does satisfy me. So the answer would be $A\delta (ct+x)+B\delta (ct-x)$ where A and B are not necessarily the constants I started with.
I hope it's right.

7. Nov 24, 2012

### fluidistic

I just found out the solution to the problem. Apparently A and B depend on k (http://mathworld.wolfram.com/WaveEquation1-Dimensional.html), I missed this. The answer should be f(x-ct)+f(x+ct) instead of the delta. f could be any function twice differentiable I think, thus the delta is a possibility.
P.S.:I made a typo in my previous post. The argument of the second delta should be "x-ct".