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Old Dec 2nd 2010, 02:43 PM   #1
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Fourier Transforms

First let me apologize if I'm posting this in the wrong place. I'm new to the forums, and would love to be a normal participant in discussions, but have to get started somewhere. Also forgive me as I am not experienced in using mathematics in forums other than what you will see posted here. If there are better ways, please let me know. Thank you


I'm studying for a test that I have tomorrow, and one of our homework assignments over the past couple of weeks was from Essential Mathematical Methods for Physicists; Weber and Arfken. (This is the alternative edition.)

Pg 694.
# 15.2.4 Find the Fourier sine and cosine transforms of e^(−a|t|)

I know the Fourier sine and cosine transforms are as follows
Fs(ω) = Sqrt(2/π) Integrate[f(t)sin(ωt), 0, infinity]dt
Fc(ω) = Sqrt(2/π) Integrate[f(t)cos(ωt), 0, infinity]dt

I think the issue that I am having is simply how to address the exp[-a|t|] where |t| is abs[t] or the "absolute value of t."

I'm not sure how to start the integral for e^(−a|t|) sin(ωt) and e^(−a|t|) cos(ωt).

Might I be better off converting sine and cosine to their exponential forms?

Thanks in advance.
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Old Dec 2nd 2010, 03:10 PM   #2
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Something my professor mentioned was that for integrating e^(−a|t|) sin(ωt) and e^(−a|t|) cos(ωt) you end up with

integral of e^(−a|t|) sin(ωt) from 0 to infinity =

ω / (a^2 + ω^2)

and

integral of e^(−a|t|) cos(ωt) from 0 to infinity =

a / (a^2 + ω^2)

I'm not sure how she got these though.
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