Fourier transfrom of exponenetial function

[panda]

i saw the tranformation pair of the even function like

f(x) = exp(-a|x|)

have u ever seen the transformation of expo.

f(x) = exp(-ax) , where x >= 0

thans a lot!

 

[me2please]

:sm31: Examples are not for memorizing. They are used to demonstrate how to apply the definition.

Use definition!

\[F(\omega) = \int\limits_{-\infty}^{\infty}f(t)e^{-j \omega t}dt\]

\[ \int\limits_{-\infty}^{\infty}|f(t)|dt < \infty \] for Fourier to exists

Then, for a>0, Fourier transform of \[ f(t)=e^{-at}, t\geq 0 \; is \]
\[F(\omega) =\int\limits_{0}^{\infty} e^{-at}e^{-j \omega t}dt \\ =\int\limits_{0}^{\infty}e^{-(a+j \omega )t}dt \\ =\frac{-1}{a+j\omega }e^{-(a+j \omega )t} \; |_{t=0}^{\infty} \\ =\frac{1}{a+j \omega} \; , a>0 \]