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.
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
$\displaystyle \displaystyle{U(0,T)=\exp\left(-\frac{T^{2}}{2T_{0}^{2}}\right),}$ but also
$\displaystyle \displaystyle{\tilde{U}(0,\omega)=\int_{-\infty}^{\infty}U(0,T)\exp(i\omega T)\,dT.}$
Taken together, you've got
$\displaystyle \displaystyle{\tilde{U}(0,\omega)=\int_{-\infty}^{\infty}\exp\left(-\frac{T^{2}}{2T_{0}^{2}}\right)\exp(i\omega T)\,dT.}$
All I've done so far is plug $\displaystyle U(0,T)$ into the expression for $\displaystyle \tilde{U}(0,\omega),$ right?
Now, this is a doable integral as is. Do you know how to integrate Gaussian integrals?