I am unsure how to solve an integral equation. As you know the orthogonality relation for cylindrical harmonics is: $$ \int_0^{2\pi}e^{in\phi'}e^{-im\phi'}d\phi'=2\pi\delta_{m,n}\ $$
The problem I have comes when you add a unit vector inside the integral:
$$ \int_0^{2\pi}\hat{\phi'}e^{in\phi'}e^{-im\phi'}d\phi' $$
How can this integral be solved?