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Where [itex]0<r<a,\;0<\theta<\pi,\;0<\phi<2\pi,\;t>0[/itex]

with Boundary condition [itex]u(a,\theta,\phi,t)=0[/itex] and initial condition [itex]u(r,\theta,\phi,0)=f(r,\theta,\phi)[/itex].

I understand how to get to

[tex]u(r,\theta,\phi,t)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} B_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}[/tex]

What I don't understand is the next step, the book assume

[tex]g(r,\theta,\phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n} g_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta, \phi)e^{-\lambda^2_{n,j} t}[/tex]

Where [itex]g_{jnm}[/itex] is another constant.

How do you justify to assume [itex]g(r,\theta,\phi)[/itex]?