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\)
Section 22.13 Legendre Polynomial Series
There is a very powerful mathematical theorem which says that any sufficiently smooth function \(f(z)\text{,}\) defined on the interval \(-1<z<1\text{,}\) can be expanded as a linear combination of Legendre polynomials
\begin{equation}
f(z) = \sum_{\ell=0}^\infty c_\ell \, P_\ell(z)\tag{22.13.1}
\end{equation}
(This theorem is the analogue of the theorem which says that any sufficiently smooth periodic function can be expanded in a Fourier series.) You will have several occasions in physics to expand functions in Legendre polynomial series, so we will explore the technique in this section.
We can find the coefficients
\(c_\ell\) by taking the inner product of both sides of
(22.13.1) in turn with each “basis vector”
\(P_\ell\) and using
(22.12.11) . This yields
\begin{align}
\int\limits_{-1}^1 P^*_k(z) \, f(z) \, dz
\amp= \int\limits_{-1}^1 P^*_k(z)
\sum_{\ell=0}^\infty c_\ell \, P_\ell(z) \, dz\notag\\
\amp= \sum_{\ell=0}^\infty \, c_\ell
\int\limits_{-1}^1 P^*_k(z) \, P_\ell(z) \, dz\notag\\
\amp= \sum_{\ell=0}^\infty \,
c_\ell \, \frac{\delta_{k\ell}}{\ell+\frac12}\notag\\
\amp= \frac{c_k}{k+\frac12}\tag{22.13.2}
\end{align}
or equivalently
\begin{equation}
c_k = \left(k+\frac12\right) \int\limits_{-1}^1 P^*_k(z) \, f(z) \, dz\tag{22.13.3}
\end{equation}
This expression should be compared with the exponential version of a Fourier series for \(f(z)\) on the same interval \(-1\le z\le1\text{,}\) namely
\begin{equation}
f(z) = \sum_{n=-\infty}^{\infty} C_n \, e^{in\pi z}\tag{22.13.4}
\end{equation}
where
\begin{equation}
C_n = \frac1{2} \int\limits_{-1}^1 e^{-in\pi z} f(z) \, dz\tag{22.13.5}
\end{equation}
Note the analogous role played by the
normalization constants \(k+\frac12\) and
\(\frac1{2}\text{.}\) If we had made an unconventional, but more convenient, choice for the normalization for the Legendre polynomials such that the value of the integrals in
(22.12.11) were simply
\(\delta_{k\ell}\text{,}\) then we would not need to carry around the extra factor of
\(k+\frac12\) in
(22.13.3) .
Example 22.3 . Example: Legendre Expansion of \(\varepsilon(z)\) .
Consider the step function
\begin{equation}
\varepsilon(z)
= 2\,\Theta(z) - 1
= \begin{cases}+1\amp(z>0)\\-1\amp(z<0)\end{cases}\tag{22.13.6}
\end{equation}
where
\(\Theta\) is the Heaviside step function; note that
\(\varepsilon(z)\) is an
odd function of
\(z\text{.}\) Using
(22.13.3) leads to
\begin{align}
c_\ell
\amp= \left(\ell+\frac12\right)
\int\limits_{-1}^1 P^*_\ell(z) \, \varepsilon(z) \, dz\notag\\
\amp= - \left(\ell+\frac12\right)
\int\limits_{-1}^0 P^*_\ell(z) \, dz
+ \left(\ell+\frac12\right)
{\int\limits}_0^1 P^*_\ell(z)\, dz\tag{22.13.7}
\end{align}
and each integral in the final expression is an elementary integral of a polynomial. Furthermore, it is easily seen that these two integrals cancel if \(\ell\) is even, and add if \(\ell\) is odd, so that
\begin{equation}
c_\ell
= \begin{cases}
0 \amp (\ell~\text{even}) \\
\displaystyle
2\left(\ell+\frac12\right) {\int\limits}_0^1 P^*_\ell(z)\, dz
\amp (\ell~\text{odd})
\end{cases}\tag{22.13.8}
\end{equation}
These coefficients are easily evaluated on Maple for as many values of \(\ell\) as desired.
