Solve: $\int_{0}^{2\pi}g \psi e^{i n \theta}\,\text{d}\theta = n/(n-i\alpha) \int_{0}^{2\pi}\psi e^{i n \theta}\,\text{d}\theta$

Question

For $\alpha>0$, I want to find a $g(\alpha, \theta)$ such that

$$ \int_{0}^{2\pi}g(\alpha, \theta)\psi(\theta)e^{i n \theta}\,\text{d}\theta = \frac{n}{n-i\alpha} \int_{0}^{2\pi}\psi(\theta)e^{i n \theta}\,\text{d}\theta $$

for some $\psi(\theta)$. Note that neither $g$ nor $\psi$ are functions of $n$. I don't even know if this is possible for a general $\psi$, but I would like to know if anyone has some ideas.