- #1
jtleafs33
- 28
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Homework Statement
Find the Fourier Transform of:
f(t)=[itex]\frac{cos(\alpha t)}{t^2+\beta^2}[/itex]
Homework Equations
F[itex](\omega)[/itex]=[itex]\frac{1}{2\pi}[/itex][itex]\int[/itex][itex]^{∞}_{-∞}[/itex][itex]\frac{cos(\alpha t)exp(i \omega t)}{t^2+\beta^2}[/itex]
The Attempt at a Solution
I start with:
[itex]cos(\alpha t)[/itex]=[itex]\frac{exp(i \alpha t)+exp(-i \alpha t)}{2}[/itex]
Substituting this in, I get:
[itex]F(\omega)[/itex]=[itex]\frac{1}{4\pi}\int^{∞}_{-∞}\frac{exp(it(\alpha+\omega))+exp(it(\omega-\alpha))}{t^2+\beta^2}[/itex]
From here I know I should be able to get this in the form of some delta functions but I can't figure out the manipulation. I'd appreciate any help!