I have tried to evaluate the following integral for the last few hours, and I did not succeed:
$ \int\limits_{0}^{2 \pi} e^{\mathrm{i} \cdot n \cdot\mathrm{arcsin}(r \cdot\mathrm{sin}(\theta))} \cdot e^{\mathrm{i}\cdot m \cdot \mathrm{arcsin}(r \cdot \mathrm{cos}(\theta))} d \; \theta$
for $0
$ \int\limits_{0}^{2 \pi} e^{\mathrm{i} \cdot n \cdot\mathrm{arctan}(t \cdot\mathrm{sin}(\theta))} \cdot e^{\mathrm{i}\cdot m \cdot \mathrm{arctan}(t \cdot \mathrm{cos}(\theta))} d \; \theta.$
Here $m$ and $n$ are integers, and $t \in \mathbb{R}$ is scalar.
I am pretty sure that is nonzero, if and only if $n=m$, and indepedent of $r$ otherwise, but I cannot figure what substitution makes this easy to see.