# Fourier cosine and sine transform of $\exp{(-ax)}(1+bx)^{-1}$ and $\exp{(-ax)}(1+bx)^{-2}$

As stated in the title I should calculate the cosine and sine Fourier transform of:

$$f_1(x)=\exp{(-ax)}(1+bx)^{-1}$$

and

$$f_2(x)=\exp{(-ax)}(1+bx)^{-2}$$

That obviously means calculating:

$$\int_0^{\infty}\, f_i(x)\cos(\omega x)dx$$ and $$\int_0^{\infty}\, f_i(x)\sin(\omega x)dx$$

Are those definite integrals known?

By considering the sine and cosine functions as the imaginary and real part of a complex exponential, the problem boils down to finding: $$I_1(z) = \int_{0}^{+\infty}\frac{e^{-zx}}{1+x}\,dx, \qquad I_2(z) = \int_{0}^{+\infty}\frac{e^{-zx}}{(1+x)^2}\,dx,$$ (where $\text{Re}(z)>0$) both depending on the incomplete $\Gamma$ function: $$I_1(z) = e^{z}\, \Gamma(0,z),\qquad I_2 = 1-z\,e^z\, \Gamma(0,z).$$
• $I_1(z)$ and $I_2(z)$ are the wanted Fourier series, so what else do you need? To get rid of the two parameters, just set $x=\frac{1}{b}x'$. – Jack D'Aurizio Feb 2 '15 at 18:14