Computing Integrals with Complex Analysis

[babyrudin]

Homework Equations

Using complex analysis, compute
$$\int_{-\infty}^{\infty} \frac{e^{itx}}{1+x^2}dx$$
where t is real.

The Attempt at a Solution

I'm not good at complex analysis at all and am totally lost. I do know some Fourier analysis though and using it I got
$$\pi e^{-|t|}$$.
How should I solve it using complex analysis?

 

[quantumdude]

Try using the Residue Theorem.

 

[babyrudin]

Great, I think I know how to do it now. I was trying the Cauchy integral formula too much. Thanks!