Having trouble with the following finding the fourier integral representation for the following:
f(t) = te^-|t|
Ive got answer but I need verfication
I've been working on a similar problem myself, i've plotted the graph in MATLAB and see that it is an odd function, equating all A terms to 0.
I'm having trouble doing the integral of t*exp(-abs(t))sin(wt) thought. Any help would be greatly appreciated!!
You will have to sort out the constants and a few other things yourself, but we are interested in:
$\displaystyle F(\omega)=\int_{-\infty}^{\infty} t e^{|t|}e^{i \omega t} ~dt$
Split the integral into two parts:
$\displaystyle F(\omega)=\int_{0}^{\infty} t e^{t}e^{i \omega t} ~dt
+\int_{-\infty}^{0} t e^{-t}e^{i \omega t} ~dt$
in the second integral change the variable $\displaystyle \tau=-t$:
$\displaystyle F(\omega)=\int_{0}^{\infty} t e^{t}e^{i \omega t} ~dt
+\int_{\infty}^{0} \tau e^{\tau}e^{-i \omega \tau} ~d \tau$
Now the second integral is minus the complex conjugate of the first so we can write:
$\displaystyle F(\omega)=2 i {\rm{Im}} \left[ \int_{0}^{\infty} t e^{t}e^{i \omega t} ~dt\right]=
2 i {\rm{Im}} \left[ \int_{0}^{\infty} t e^{(1+i \omega) t} ~dt\right]
$
and the integral inside the rightmost brackets can be done by parts.
RonL