Math Help - Solution Using Fourier Transform.

1. Solution Using Fourier Transform.

Hi all.
Plz find attached the word file. Can anyone help me in finding how can I get the equation 4 i.e. solution by using equations 1,2 and 3 ?
Thanks.

2. Word files are generally frowned upon in this forum. A lot of members won't open Word files because of possible viruses in macros. Can you convert to pdf?

3. Dear Ackbeet,
Plz find the pdf here.
Thanks.

4. Thanks for posting a pdf. I have a question: what is the question? What are you being asked to do here? There doesn't seem to be any sort of request here. Prove equation (4)?

5. Dear Ackbeet.
You are right. Equation 4 needs to be proved.
Thanx.

6. Very well, then. It looks like a straight-forward integration problem, then. You've got to plug things into things, and then plug those into things. So, for example, you have

$\displaystyle{U(0,T)=\exp\left(-\frac{T^{2}}{2T_{0}^{2}}\right),}$ but also

$\displaystyle{\tilde{U}(0,\omega)=\int_{-\infty}^{\infty}U(0,T)\exp(i\omega T)\,dT.}$

Taken together, you've got

$\displaystyle{\tilde{U}(0,\omega)=\int_{-\infty}^{\infty}\exp\left(-\frac{T^{2}}{2T_{0}^{2}}\right)\exp(i\omega T)\,dT.}$

All I've done so far is plug $U(0,T)$ into the expression for $\tilde{U}(0,\omega),$ right?

Now, this is a doable integral as is. Do you know how to integrate Gaussian integrals?