Fourier transfrom af exponenetial funktion

[panda]

Jeg så tranformation par de selv fungerer som

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

have u nogensinde set den forvandling af EXPO.

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

thans a lot!

 

[me2please]

: sm31: Eksempler er ikke for Lagring.De bruges til at demonstrere, hvordan man kan anvende den definition.Brug definition!
<img src='http://www.elektroda.pl/cgi-bin/mimetex/mimetex.cgi?3$F(\omega) = \int\limits_{-\infty}^{\infty}f(t)e^{-j \omega t}dt' title="3 $ F (\ omega) = \ int \ limits_ (- \ infty) ^ (\ infty) f (t) e ^ (-j \ omega t) dt" alt='3$F(\omega) = \int\limits_{-\infty}^{\infty}f(t)e^{-j \omega t}dt' align=absmiddle>

<img src='http://www.elektroda.pl/cgi-bin/mimetex/mimetex.cgi?3$ \int\limits_{-\infty}^{\infty}|f(t)|dt < \infty ' title="3 $ \ int \ limits_ (- \ infty) ^ (\ infty) | f (t) | dt <\ infty" alt='3$ \int\limits_{-\infty}^{\infty}|f(t)|dt < \infty ' align=absmiddle>

til Fourier at der findes

Så, for a> 0, Fouriertransformation af

<img src='http://www.elektroda.pl/cgi-bin/mimetex/mimetex.cgi?3$ f(t)=e^{-at}, t\geq 0 \; is ' title="3 $ f (t) = e ^ (-at), t \ geq 0 \; er" alt='3$ f(t)=e^{-at}, t\geq 0 \; is ' align=absmiddle>
<img src='http://www.elektroda.pl/cgi-bin/mimetex/mimetex.cgi?3$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 ' title="3 $ 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" alt='3$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 ' align=absmiddle>