Given a set of functions $f_{mv}(r,\phi)=J_{v}(k_{mv}r)\cos(v \phi)$ in polar coordinates, where $m$ denotes the mth root of $J'_{v}(k_{mv}a)=0$ at boundary $r=a$, how do we prove that these functions are orthogonal iif $v$ is integer?
And then, for a case where we have rational $v$ (ratio of two integers, $v=M/N$), is there a way to construct a new orthogonal set that is of similar form to the one above (or linearly re-adjust the original one above to make it orthogonal under such condition)?