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-rw-r--r--buch/papers/kugel/spherical-harmonics.tex23
1 files changed, 13 insertions, 10 deletions
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 0fb6557..49b9c06 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -322,7 +322,8 @@ obtain the \emph{associated Legendre functions}.
The functions
\begin{equation}
P^m_n (z) = (1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z)
- = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n, \quad |m|<n
+ = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}
+ \frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n, \quad |m|<n
\end{equation}
are known as Ferrers or associated Legendre functions.
\end{definition}
@@ -561,6 +562,13 @@ product:
\int_0^{2\pi} e^{i(m - m')\varphi}
\, d\varphi.
\end{align*}
+ Essentially, what we just did was to turn
+ \eqref{kugel:eq:spherical-harmonics-inner-prod} in this form:
+ \(
+ \langle Y^m_n, Y^{m'}_{n'} \rangle_{\partial S}
+ = \langle P^m_n, P^{m'}_{n'} \rangle_z
+ \; \langle e^{im\varphi}, e^{-im'\varphi} \rangle_\varphi
+ \).
First, notice that the associated Legendre polynomials are assumed to be real,
and are thus unaffected by the complex conjugation. Then, we can see that when
$m = m'$ the inner integral simplifies to $\int_0^{2\pi} 1 \, d\varphi$ which
@@ -574,13 +582,8 @@ product:
\end{equation*}
where in the second step we performed the substitution $z = \cos\vartheta$;
$d\vartheta = \frac{d\vartheta}{dz} dz= - dz / \sin \vartheta$, and then we
- used lemma \ref{kugel:thm:associated-legendre-ortho}.
- We are allowed to use
- the lemma because $m = m'$. After the just mentioned substitution we can write eq.\eqref{kugel:eq:spherical-harmonics-inner-prod} in this form
- \begin{equation*}
- \langle Y^m_n, Y^{m'}_{n'} \rangle_{\partial S} = \langle P^m_n, P^{m'}_{n'} \rangle_z \; \langle e^{im\varphi}, e^{-im'\varphi} \rangle_\varphi.
- \end{equation*}
- Now we just need look at the case when $m \neq m'$. Fortunately this is
+ used lemma \ref{kugel:thm:associated-legendre-ortho}. We are allowed to use
+ the lemma because $m = m'$. Now we just need look at the case when $m \neq m'$. Fortunately this is
easier: the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$,
or in other words we are integrating a complex exponential over the entire
period, which always results in zero. Thus, we do not need to do anything and
@@ -654,8 +657,8 @@ At this point we have shown that the spherical harmonics form an orthogonal
system, but in many applications we usually also want a normalization of some
kind. For example the most obvious desirable property could be for the spherical
harmonics to be ortho\emph{normal}, by which we mean that $\langle Y^m_n,
-Y^{m'}_{n'} \rangle = \delta_{nn'}$. To obtain orthonormality, we simply add an
-ugly normalization factor in front of the previous definition
+Y^{m'}_{n'} \rangle = \delta_{nn'} \delta_{mm'}$. To obtain orthonormality, we
+simply add an ugly normalization factor in front of the previous definition
\ref{kugel:def:spherical-harmonics} as follows.
\begin{definition}[Orthonormal spherical harmonics]