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| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 15 |
1 files changed, 8 insertions, 7 deletions
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex index fb5a144..3100e36 100644 --- a/buch/papers/kugel/spherical-harmonics.tex +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -311,10 +311,11 @@ such that they also become solutions of the associated Legendre equation What is happening in lemma \ref{kugel:thm:extend-legendre}, is that we are essentially inserting a square root function in the solution in order to be able to reach the parts of the domain near the poles at $\pm 1$ of the associated -Legendre equation, which is not possible only using power series -\kugeltodo{Reference book theory on extended power series method.}. Now, since -we have a solution in our domain, namely $P_n(z)$, we can insert it in the lemma -obtain the \emph{associated Legendre functions}. +Legendre equation, which is not possible only using power series (see sections +\ref{buch:differentialgleichungen:section:potenzreihenmethode} and +\ref{buch:differentialgleichungen:subsection:verallgemeinrt} for a discussion). +Now, since we have a solution in our domain, namely $P_n(z)$, we can insert it +in the lemma obtain the \emph{associated Legendre functions}. \begin{definition}[Ferrers or associated Legendre functions] \label{kugel:def:ferrers-functions} @@ -595,9 +596,9 @@ These proofs for the various orthogonality relations were quite long and algebraically tedious, mainly because they are ``low level'', by which we mean that they (arguably) do not rely on very abstract theory. However, if we allow ourselves to use the more abstract Sturm Liouville theory discussed in chapters -\ref{buch:integrale:subsection:sturm-liouville-problem} and \kugeltodo{reference -to chapter 17 of haddouche and Löffler} the proofs can become ridiculously -short. Let's do for example lemma \ref{kugel:thm:associated-legendre-ortho}. +\ref{buch:integrale:subsection:sturm-liouville-problem} and +\ref{chapter:sturmliouville} the proofs can become ridiculously short. Let's do +for example lemma \ref{kugel:thm:associated-legendre-ortho}. \begin{proof}[ Shorter proof of lemma \ref{kugel:thm:associated-legendre-ortho} |