Show that the substitution \(u=R(\rho) \Phi(\phi) \Theta(\theta)\) in the
problem in spherical coordinates for the domain \(\rho<1\) :
$$
\begin{gathered}
\nabla^{2} u+\lambda u \equiv \frac{1}{\rho^{2} \sin ^{2} \phi}\left[\sin ^{2}
\phi \frac{\partial}{\partial \rho}\left(\rho^{2} \frac{\partial u}{\partial
\rho}\right)+\sin \phi \frac{\partial}{\partial \phi}\left(\sin \phi
\frac{\partial u}{\partial \phi}\right)+\frac{\partial^{2} u}{\partial
\theta^{2}}\right]+\lambda u=0, \\
u(\rho, \phi, \theta)=0 \quad \text { for } \rho=1 .
\end{gathered}
$$
leads to the separate Sturm-Liouville problems:
$$
\begin{gathered}
\left(\rho^{2} R^{\prime}\right)^{\prime}+\left(\lambda \rho^{2}-\alpha\right)
R=0, \quad R=0 \quad \text { for } \rho=1 \\
\left(\sin \phi \Phi^{\prime}\right)^{\prime}+(\alpha \sin \phi-\beta \csc
\phi) \Phi=0, \quad \Theta^{\prime \prime}+\beta \Theta=0
\end{gathered}
$$
Here \(\alpha, \beta\), and \(\lambda\) are characteristic values to be
determined. The condition that \(u\) be continuous throughout the sphere
requires \(\Theta\) to have period \(2 \pi\), so that \(\beta=k^{2}(k=0\), \(1,2,
\ldots)\) and \(\Theta_{k}(\theta)\) is a linear combination of \(\cos k \theta\)
and \(\sin k \theta\). When \(\beta=k^{2}\), it can be shown that continuous
solutions of the second problem for \(0 \leq \phi \leq \pi\) are obtainable only
when \(\alpha=n(n+1), k=0,1, \ldots, n\), and \(\Phi\) is a constant times \(P_{n,
k}(\cos \phi)\), where
$$
P_{n, k}(x)=\left(1-x^{2}\right)^{\frac{1}{2}} \frac{d^{k}}{d x^{k}} P_{n}(x)
$$
and \(P_{n}(x)\) is the \(n\)th Legendre polynomial. When \(\alpha=n(n+1)(n=0,1,2,
\ldots)\), the first problem has a solution continuous for \(\rho=0\) only when
\(\lambda\) is one of the roots \(\lambda_{n+\lfloor 1,1},
\lambda_{n+\frac{1}{2}, 2}, \ldots\) of the function
\(J_{n+\frac{1}{!}}(\sqrt{x})\), where \(J_{n+\frac{1}{2}}(x)\) is the Bessel
function of order \(n+\frac{1}{2}\); for each such \(\lambda\) the solution is a
constant times \(\rho^{-\frac{1}{2}} J_{n+\frac{1}{2}}(\sqrt{\lambda} \rho)\).
Thus one obtains the characteristic functions \(\rho^{-\frac{1}{2}}
J_{n+\frac{1}{1}}(\sqrt{\lambda} \rho) P_{n, k}(\cos \phi) \cos k \theta,
\rho^{-\frac{1}{2}} J_{n+\frac{1}{2}}(\sqrt{\lambda} \rho) P_{n, k}\) \((\cos
\phi) \sin k \theta\), where \(n=0,1,2, \ldots, k=0,1, \ldots, n\), and \(\lambda\)
is chosen as above for each \(n\). It can be shown that these functions form a
complete orthogonal system for the spherical region \(\rho \leq 1\).
