Does anybody know how to solve the following singular integral:
$$ \int_0^\pi \frac{\text{e}^{-i\cdot a\cdot \cos(x)}}{\cos(x)-b}\,\text{d}x $$
with $x$, $a$ and $b$ being real-valued. Do integrals of this general form belong to a specific class of integrals?
The inequality constraint on b is: $$ \lvert b\rvert\leq1 $$
$\displaystyle{% {\cal I} \equiv \int_{0}^{\pi}\frac{\text{e}^{-{\rm i}a\cos\left(x\right)}} {\cos\left(x\right)-b}\,\text{d}x}$
${\large\mbox{Just a hint !!!}\,:}$ $$ {\cal I} = {\rm e}^{-{\rm i}ab} \int_{0}^{\pi}\frac{\text{e}^{-{\rm i}a\left\lbrack\cos\left(x\right) - b\right\rbrack}} {\cos\left(x\right)-b}\,\text{d}x $$
\begin{align} {\partial\left({\rm e}^{{\rm i}ab}{\cal I}\right) \over \partial a} = -{\rm i}{\rm e}^{{\rm i}ab}\int_{0}^{\pi} \text{e}^{-{\rm i}a\cos\left(x\right)}\,\text{d}x = -{\rm i}{\rm e}^{{\rm i}ab}\ \pi\,{\rm J}_{0}\left(a\right) \end{align}
${\rm J}_{0}\left(z\right)$ is a ${\bf\mbox{Bessel function of the first kind}}$.