Is there a way to define the Fourier transform of the function $e^{in\theta}$ on the 2D plane in polar coordinates, with $n$ an integer? $$\int d^2x e^{i \vec{k}\cdot \vec{x}}e^{in\theta}=F(\vec{k})?\qquad \vec{x}=(r\cos\theta, r\sin\theta)$$ Clearly $e^{in\theta}$ is ill-defined at $\vec{x}=0$, but I was hoping there might be a way to make sense of this integral, perhaps considering $F$ as a generalized function.
Here's what I've tried: $$\vec{k}=(k\cos\phi,\,k\sin\phi)$$ $$\int d^2x e^{i \vec{k}\cdot \vec{x}}e^{in\theta}=\int dr\,r\int d \theta e^{ikr\cos(\theta-\phi)+in\theta}=2\pi e^{in\phi}\int_0^\infty dr \,r J_n(kr)$$ The integral over $r$ doesn't converge. For instance, when $n=0$, we can say $$\int^\Lambda_0dr \,r J_0(kr)=\frac{\Lambda}{k}J_1(k\Lambda)\sim \frac{\sqrt{\Lambda}}{k^{3/2}}$$ using $J_1(r)\sim r^{-1/2}$ for large $r$. So there isn't an obvious limit when $\Lambda\rightarrow \infty$.
However we know that when $n=0$ this does have a Fourier transform in terms of a delta function. So that's why I am hoping there are also useful generalized functions when $n\neq 0$.
I say 'useful' because I see by the convolution theorem that if $\tilde{f}(k)$ is the Fourier transform of some $f(x)$ then $\int \frac{d^2k}{(2\pi)^2}\tilde{f}(k) F(q-k)$ is the Fourier transform of $f(x)e^{in\theta}$ evaluated at $q$. But I am wondering if there is some description of this purely in terms of momentum space.