# Fourier integral representation

• May 28th 2008, 02:27 AM
taltas
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
• May 28th 2008, 05:45 AM
mr fantastic
Quote:

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 .....
• May 28th 2008, 09:54 AM
bcvw85
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!!
• May 28th 2008, 11:45 AM
CaptainBlack
Quote:

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