Show that the following integral is well defined:

$$I(a, b)=\int_{0}^{\infty}\left(\frac{e^{-b x}}{e^{i a} e^{x}-1}-\frac{e^{b x}}{e^{-i a} e^{x}-1}\right) d x, \quad 0<a<\infty, a \neq 2 n \pi, n \in \mathbb{Z}, 0<b<1$$

Express $I(a, b)$ in terms of a combination of hypergeometric functions.

[You may assume without proof that the hypergeometric function $F(a, b ; c ; z)$ can be expressed in the form

$$F(a, b ; c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

for appropriate restrictions on $c, b, z$. Furthermore,

$$\Gamma(z+1)=z \Gamma(z) \cdot]$$