1. ## Fourier integral representation

Having trouble with the following finding the fourier integral representation for the following:

f(t) = te^-|t|

Ive got answer but I need verfication

2. Originally Posted by taltas
Having trouble with the following finding the fourier integral representation for the following:

f(t) = te^-|t|

Ive got answer but I need verfication
Showing your working and answer might expedite the process of helping you. You did realise that f(t) is odd, yes? That simplifies the calculation .....

3. 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!!

4. Originally Posted by taltas
Having trouble with the following finding the fourier integral representation for the following:

f(t) = te^-|t|

Ive got answer but I need verfication
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