Change of Variables

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Section 15.2 Change of Variables

Since we have solved the \(\phi\) equation (6.5.18) and found the possible values of the separation constant \(\sqrt{B}=m\in\{0,\pm1,\pm2,...\}\text{,}\) the \(\theta\) equation becomes an eigenvalue/eigenfunction equation for the unknown separation constant \(A\) and the unknown function \(P(\theta)\text{.}\)

\begin{equation} \left({\sin\theta}\frac{d}{d\theta} \left({\sin\theta}\frac{d}{d\theta}\right) - A\sin^2\theta - m^2\right) P(\theta) = 0\tag{15.2.1} \end{equation}

Figure 2. Relationship between \(z\) and \(\theta\text{.}\)

We start with a change of independent variable \(z=\cos\theta\) where \(z\) is the usual rectangular coordinate in three-space. As \(\theta\) ranges from \(0\) to \(\pi\text{,}\) \(z\) ranges from \(1\) to \(-1\text{.}\) We see from Figure 2 that:

\begin{equation} \sqrt{1-z^2}=\sin\theta\tag{15.2.2} \end{equation}

Using the chain rule for partial derivatives, we have:

\begin{equation} \frac{d}{d\theta} = \frac{d z}{d\theta}\frac{d}{d z} = -\sin\theta\frac{d}{d z} = -\sqrt{1-z^2}\frac{d}{d z}\tag{15.2.3} \end{equation}

Notice, particularly, the last equality: we are trying to change variables from \(\theta\) to \(z\text{,}\) so it is important to make sure we change all the \(\theta\)’s to \(z\)’s. Multiplying by \(\sin\theta\) we obtain:

\begin{equation} \sin\theta\frac{d}{d\theta} = -\left(1-z^2\right)\frac{d}{d z}\tag{15.2.4} \end{equation}

Be careful finding the second derivative; it involves a product rule:

\begin{align} {\sin\theta}\frac{d}{d\theta}\left( {\sin\theta}\frac{d}{d\theta}\right) \amp= \left(1-z^2\right)\frac{d}{d z} \left(\left(1-z^2\right)\frac{d}{d z}\right)\notag\\ \amp= \left(1-z^2\right)^2\frac{d^2}{d z^2} - 2z\left(1-z^2\right)\frac{d}{d z}\tag{15.2.5} \end{align}

Inserting (15.2.2) and (15.2.5) into (15.2.1), we obtain a standard form of the Associated Legendre’s equation:

\begin{equation} \frac{d^2 P}{d z^2} - \frac{2z}{1-z^2}\frac{d P}{d z} - \frac{A}{1-z^2} P-\frac{m^2}{(1-z^2)^2} P = 0\tag{15.2.6} \end{equation}

In Section 7.2 and Section 15.3, we will solve this equation. After we have found the eigenfunctions \(P(z)\text{,}\) we will substitute \(z=\cos\theta\) everywhere to find the eigenfunctions of the original equation (15.2.1).