Fourier Transform of $f(x) = \sin(x) \cdot e^{-|x|}$

Im trying to find the Fourier Transform of $f(x) = \sin(x) \cdot e^{-|x|}$

I applied the standard formula and got to this point:

$$\tilde{f(t)}= \frac{1}{\sqrt{2\pi}}\cdot \int_{-\infty}^{+\infty}\sin(x)e^{-|x|}e^{-itx} dx$$

= $$\frac{1}{\sqrt{2\pi}}\cdot \int_{-\infty}^{0}\sin(x)e^{x}e^{-itx} dx+\frac{1}{\sqrt{2\pi}}\cdot \int_{0}^{+\infty}\sin(x)e^{-x}e^{-itx} dx$$

How can I integrate this - I tried integration by parts and did not get anywhere because there will always be alternation between sin and cos and also there is an e function in the integrand.

Now Im trying to think about integrating by the help of substitution but I couldn't think of anything to substitute it with. Can you please help :) ?

Thanks a lot!

• use that $sin(x) = \frac {e^{i\psi t} - e^{-i \psi t}}2$ – mike Apr 30 '12 at 20:03

In general, if you use integration by parts twice, you can solve it like this: $$I=\int{\sin(x)e^{ax}dx}=-\cos(x)e^{ax} + a\int{\cos(x)e^{ax}dx} = -\cos(x)e^{ax} + a\sin(x)e^{ax} - a^2\int{\sin(x)e^{ax}dx}= -\cos(x)e^{ax} + a\sin(x)e^{ax} -a^2I$$
$$I = \frac{-\cos(x)e^{ax} + a\sin(x)e^{ax}}{1+a^2}$$