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diff --git a/notes/FourierOnS2.tex b/notes/FourierOnS2.tex index 24b972f..edd4813 100644 --- a/notes/FourierOnS2.tex +++ b/notes/FourierOnS2.tex @@ -227,7 +227,7 @@ Finding the solutions to this equation is so involved, that it deserves its own \subsection{The associated Legendre polynomials} -In this section we would like to find the solutions to the \emph{associated} Legendre equation, which is actually a generalization of Legendre equation: +In this section we would like to find the solutions to the \emph{associated} Legendre equation, which is actually a generalization of Legendre's equation: \begin{equation} \label{eqn:legendre} \left( 1 - x^2 \right) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + n(n + 1) y(x) = 0. @@ -240,15 +240,40 @@ Thus we first need examine the solutions to this equation before constructing th P_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{(-1)^k (2n-2k)!}{2^n k! (n-k)!(n-2k)!} x^{n-2k}, \end{equation} - known as Legendre's polynomials are solutions to Legendre's equation \eqref{eqn:legendre}. + are solutions to Legendre's equation \eqref{eqn:legendre} when \(n > 0\). \end{proposition} -\begin{lemma} The expression +The proof for this proposition is quite algebraically involved and is thus left in the appendix. Since this is a power series \eqref{eqn:legendre-poly} can also be rewritten using Gauss' Hypergeometric function. + +\begin{proposition} + The polynomial \eqref{eqn:legendre-poly} can we rewritten using Gauss' Hypergeometric function + \[ + {}_2F_1 \left( \begin{matrix} + a_1, & a_2 \\ \multicolumn{2}{c}{b} + \end{matrix} ; \frac{1 - x}{2} \right) + = + \sum_{k = 0}^\infty \frac{(a_1)_k (a_2)_k}{(b)_k} \frac{x^k}{k!}, + \] + where the notation \((a)_k\) is for the Pochhammer Symbol + \[ + (a)_k = a (a + 1) \ldots (a + k - 1). + \] + Hence for \(x \in (-1, 1)\) and \(n \in \mathbb{R}\): + \[ + P_n (x) = {}_2F_1 \left( \begin{matrix} + n + 1, & -n \\ \multicolumn{2}{c}{1} + \end{matrix} ; \frac{1 - x}{2} \right). + \] +\end{proposition} + +In some applications, such as in quantum mechanics, it is more common to see it written yet in another form using Rodrigues' Formula. + +\begin{proposition} The expression \begin{equation} \label{eqn:legendre-rodrigues} P_n(x) = \frac{1}{n!2^n}\frac{d^n}{dx^n}(x^2-1)^n. \end{equation} is equivalent to \eqref{eqn:legendre-poly}. -\end{lemma} +\end{proposition} \begin{proof} We start expanding the term \((x^2-1)^n\); According to the binomial theorem diff --git a/notes/build/FourierOnS2.pdf b/notes/build/FourierOnS2.pdf Binary files differindex 887616d..d40262c 100644 --- a/notes/build/FourierOnS2.pdf +++ b/notes/build/FourierOnS2.pdf |