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| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 23 |
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] |