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## Homework Statement

Calculate (from the definition, no tables allowed) the Fourier Transform of [itex]e^{-a*|t|}[/itex], where a > 0.

## Homework Equations

Fourier Transform:

[itex]G(f) = \int_{-\infty}^{\infty} g(t)e^{-j\omega t} dt[/itex]

## The Attempt at a Solution

I thought I'd break up the problem into the two cases of t (where it's negative and positive). However, when I calculated the portion where t > 0, I got:

[itex]

G(f) = \int_{0}^{\infty} e^{-at} e^{-j\omega t}dt = G(f) = \int_{0}^{\infty} e^{-(j\omega + a)t}dt = \frac{e^{-(j\omega + a)t}}{-(j\omega + a)}\bigg|_0^\infty = 0 - \frac{1}{-(j\omega + a)} = \frac{1}{j\omega + a}

[/itex]

Which is nowhere close to what WolframAlpha says the answer should be:

http://www.wolframalpha.com/input/?i=fourier+transform+exp(-a*abs(t))

So I guess I'm confused on how I should even approach the problem. Any suggestions?

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