Consider a certain vector differential equation whose general solution is given by linear combination of orthogonal vector fields $$\mathbf{F}(r,\theta,\phi)=\sum_{m=0}^{\infty} \sum_{n=m}^{\infty} a_{mn} \mathbf{M}_{mn}$$
Where $\mathbf{M}$ satisfy the orthogonality relation $$ \int\mathbf{M}_{mn} \cdot \mathbf{M}_{m'n'}d\Omega=\delta_{mm'}\delta_{nn'} $$
Where the integration is over a solid angle.
It follows that $a_{mn}$ is given by $$ \int \mathbf{F} \cdot \mathbf{M}_{m'n'}d\Omega $$
Consider the following problem, we have a certain vector field $\mathbf{E}$ that satsfies the aforementioned differential equation. therefore it follows that it admits an expansion in terms of $\mathbf{M}$ $$\mathbf{E}(r,\theta,\phi)=\sum_{m=0}^{\infty} \sum_{n=m}^{\infty} b_{mn} \mathbf{M}_{mn}$$
Moreover the boundary condition is given by $$\mathbf{E}(a,\theta,\phi)=\mathbf{F}(a,\theta,\phi)$$ where the equality holds for $$ r=a, \quad 0 \le \theta \le \pi, \quad 1/2 \pi \le \phi \le 3/2\pi $$
The problem lies in finding $b_{mn}$. Now that $1/2\pi \le \phi \le 3/2\pi $, I cannot use the orthogonality relation to find $b_{mn}$. Is there a way around this?