Completeness of the functions $ exp(im\phi)$ in $[0,2\pi]$

Question

I would like to know if the set of azimuthal functions

$$\exp(im\phi), \phi \in [0,2\pi]$$

forms a complete set for $m=0,\ldots, \infty$ or in order to be complete $m$ has to run from $m=-\infty,\ldots,0,\ldots,\infty$. Thank you for any help.

Answer 1

You have include all integer values. In fact $e^{-i\phi}$ is orthogonal to $e^{mi\phi}$ for all $m \geq 0$ so $\{e^{im\phi}: m\geq 0 \}$ cannot be complete.