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-rw-r--r--buch/papers/kugel/spherical-harmonics.tex16
1 files changed, 9 insertions, 7 deletions
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 68b7eda..b46d64d 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -550,6 +550,15 @@ product:
\end{theorem}
\begin{proof}
We will begin by doing a bit of algebraic maipulaiton:
+ \footnote{
+ Essentially, what we just did was to turn
+ \eqref{kugel:eq:spherical-harmonics-inner-prod} in this form:
+ \(
+ \langle Y^m_n, Y^{m'}_{n'} \rangle_{\partial S}
+ = \langle P^m_n, P^{m'}_{n'} \rangle_z
+ \; \langle e^{im\varphi}, e^{-im'\varphi} \rangle_\varphi
+ \).
+ }
\begin{align*}
\int_{0}^\pi \int_0^{2\pi}
Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
@@ -564,13 +573,6 @@ 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