I am trying to solve a separation of variables PDE problem using Complex Fourier Series to apply the final boundary condition. So far, I have the solution $u = \sum_{-\infty}^\infty (A_n r^{-n} + B_n r^n)(C_n e^{-n\theta i} + D_n e^{n\theta i})$.
What I want and think I am algebraically allowed to do before applying the final boundary condition is drop half of each factor, to get something like $u = \sum_{-\infty}^\infty B_n r^n D_n e^{n\theta i}$. The idea is that due to the two-sided summation, each of the $A_n$ terms can be lumped into a corresponding $B_n$ term, and each of the $C_n$ terms and be lumped into a corresponding $D_n$ term (and then I can obviously lump $B_n D_n$ into a single coefficient in preparation for the Fourier Series).
Is this correct? The reason I hesitate is that if I use the same argument to arrive at, for example, $u = \sum_{-\infty}^\infty A_n r^{-n} D_n e^{n\theta i}$, the final answer should turn out different (either containing an $r^n$ or $r^{-n}$ factor).