- #1

DragonBlight

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- Homework Statement
- Finding the Fourier transform of ##f(t) = te^{-at}## if t > 0 and ##f(t) = 0## otherwise.

- Relevant Equations
- ##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} te^{-at}e^{-iwt} dt##

Doing the Fourier transform for the function above I'm getting a result, but since I can't get the function f(t) with the inverse Fourier transform, I'm wondering where I made a mistake.

##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} te^{-t(a + iw)} dt##

By integrating by part, where G = -a - iw

##F(w) = \frac{te^{Gt}}{G}|_{0}^{\infty} - \int_{0}^{\infty} \frac{e^{Gt}}{G} dt##

##= \frac{1}{G^2}##

Thus,

##F(w) = \frac{1}{\sqrt{2 \pi}} \frac{1}{(-a -iw)^2}##

##F(w) = \frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} te^{-t(a + iw)} dt##

By integrating by part, where G = -a - iw

##F(w) = \frac{te^{Gt}}{G}|_{0}^{\infty} - \int_{0}^{\infty} \frac{e^{Gt}}{G} dt##

##= \frac{1}{G^2}##

Thus,

##F(w) = \frac{1}{\sqrt{2 \pi}} \frac{1}{(-a -iw)^2}##