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

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/?i=Fourier+transform&assumption={"F",+"FourierTransformCalculator",+"transformfunction"}+->"sin(a*t)*e^(-t/b)"&assumption={"F",+"FourierTransformCalculator",+"variable1"}+->"t"&assumption={"F",+"FourierTransformCalculator",+"variable2"}+->"w"&assumption={"C",+"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

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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

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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

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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' ##.
 
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!
 
Please find attached the screenshot of WolphramAlpha's result:
242849
 

Charles Link

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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.
 
Last edited:
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.
WolframAlpha (1).png
 

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