I was looking for a generalization of the $\textit{Residue theorem}$ in the case where I consider a complex number elevated to a generic phase of the identity.
Let's do an example. Let's assume that I would like to compute
\begin{equation} \frac{1}{2\pi i}\oint_{\mid z \mid =1}\frac{dz}{z}\frac{1}{(1-\frac{t}{z})(1-tz)} \end{equation} where $t \in \mathbb{C}$ and $\mid t \mid < 1$, then the only pole inside the unit circle is located at $z=t$ and applying the $\textit{Residue theorem}$ we get \begin{equation} 2\pi i \times \frac{1}{2\pi i}\frac{1}{1-t^2} = \frac{1}{1-t^2} \end{equation} Now I'd like to consider the family of integrals (for $n>3$) \begin{equation} \frac{1}{2\pi i}\oint_{\mid z \mid =1}\frac{dz}{z}\frac{1}{(1-tz^{\omega_n})(1-tz)} \end{equation} where $\omega_n = e^{\frac{2\pi i}{n}}$ with $n \in \mathbb{Z}$. Does it exist a generalization of the $\textit{Residue theorem}$ that allow to evaluate the above integral ? Or more general, does it make sense to try to evaluate the above integral ?
Many thanks