How to calculate the Fourier transform of sin(a*t)*exp(-t/b) ?

[Leonid92]

TL;DR Summary

How to calculate Fourier transform of sin(a*t)*exp(-t/b) ?

Hi all,

I need to calculate Fourier transform of the following function: sin(a*t)*exp(-t/b), where 'a' and 'b' are constants.
I used WolphramAlpha site to find the solution, it gave the result that you can see following the link: https://www.wolframalpha.com/input/...Fourier+transform"}+->+{"Calculator",+"dflt"}
But I have doubt about this result. Could you please tell, how to calculate the Fourier transform of mentioned function manually? And is there another reliable site/program where I can find Fourier transform of any function?

 

[Charles Link]

I presume your function starts at $t=0$. First, convert $\sin(at)$ to complex form with Euler's formula. Next $\hat{F}(\omega)=\int\limits_{-\infty}^{+\infty}F(t)e^{-i \omega t} \, dt$. I believe the integrals are readily workable, where you only need to integrate from $0$ to $+\infty$.

 

[Charles Link]

And I can't get to your "link" above, but I get $\\$
$\hat{F}(\omega)=-\frac{1}{2}[\frac{(\omega-a)+ib'}{(\omega-a)^2+b'^2}-\frac{(\omega+a)+ib'}{(\omega+a)^2+b'^2} ]$ where $b'=\frac{1}{b}$.

 

[Charles Link]

The above function $\hat{F}(\omega)$ basically has peaks at $\omega=\pm a$, with a width (a spread around $\omega=\pm a$) that is approximately $b'$.

 

[Leonid92]

The above function $\hat{F}(\omega)$ basically has peaks at $\omega=\pm a$, with a width (a spread around $\omega=\pm a$) that is approximately $b'$.

Thank you very much!

 

[Leonid92]

Please find attached the screenshot of WolphramAlpha's result:

 

[Charles Link]

It's very hard to read. The result will be slightly different if you use $e^{-|t|/b}$, and let $t$ run from $-\infty$ to $+\infty$ . It looks like they get their final $\sqrt{\pi}$ in the numerator, but it's much too hard to see... They define their F.T. slightly differently...Even with a magnifying glass I can not make out their complete result.

 

[Leonid92]

It's very hard to read. The result will be slightly different if you use $e^{-|t|/b}$, and let $t$ run from $-\infty$ to $+\infty$ . It looks like they get their final $\sqrt{\pi}$ in the numerator, but it's much too hard to see... They define their F.T. slightly differently...Even with a magnifying glass I can not make out their complete result.

Sorry for bad quality of image. Please find attached the image with better quality.