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

Question

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?

Answer 1

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: $$ I_1(z) = e^{z}\, \Gamma(0,z),\qquad I_2 = 1-z\,e^z\, \Gamma(0,z). $$