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| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 16 |
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 |