Actually I would like to develop function $g(r,\phi)$ on $[0,\infty)\times[0,2\pi]$ in a complete set of functions. This does not seem to very difficult:
$$g(r,\phi) = \sum_{m=-\infty}^{\infty} R_m(r)e^{im\phi}$$
I have to make choice on the complete function set for $R_m(r)$. I run here into a problem: For instance I could develop $R_m(r)$ into Laguerre functions:
$$R_m(r) = \sum_{q=0}^{\infty} a^m_q r^{m/2}e^{-r/2} L^m_q(r)$$
I know from these functions that they form a complete set for $m\geq 0$, however I wonder which development can I choose for $R_m(r)$ if $m<0$. As far as I know the Laguerre functions $L^\alpha_q$ only exist for $\alpha>-1$. A tentative would simply be:
$$R_m(r) = \sum_{q=0}^{\infty} a^m_q r^{|m|/2}e^{-r/2} L^{|m|}_q(r)$$
but actually I don't know if this possible as it would make the modes development in $r$ the same for $m$ and $-m$. On the other hand allow for $m<0$ would mean that at $r=0$ the development $R_m(r) = \sum_{q=0}^{\infty} a^m_q r^{m/2}e^{-r/2} L^m_q(r)$ would have a singular point. So what is the proper solution to this problem ?