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

f(t) = te^-|t|

Ive got answer but I need verfication

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- May 28th 2008, 02:27 AMtaltasFourier 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 AMmr fantastic
- May 28th 2008, 09:54 AMbcvw85
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 AMCaptainBlack
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