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diff --git a/notes/FourierOnS2.tex b/notes/FourierOnS2.tex index edd4813..ced6b8d 100644 --- a/notes/FourierOnS2.tex +++ b/notes/FourierOnS2.tex @@ -7,6 +7,7 @@ %% styling for this document \usepackage{tex/docstyle} +\usepackage[firstpageonly, color={[gray]{0.9}}]{draftwatermark} %% Theorems, TODO: move to docstyle \usepackage{amsthm} @@ -101,6 +102,7 @@ The answer has to do with solving other problems. That is because originally Fou + \frac{\partial^2 f}{\partial \nu^2} = 0. \end{equation} +\texttt{[FIXME: Should be an eigenvalue problem \(\nabla_2^2 f = \lambda f\)]} This PDE is known as Laplace's equation, and can be solved by separation using the ansatz \[ f(\mu, \nu) = M(\mu)N(\nu), @@ -242,6 +244,7 @@ Thus we first need examine the solutions to this equation before constructing th \end{equation} are solutions to Legendre's equation \eqref{eqn:legendre} when \(n > 0\). \end{proposition} +\begin{proof} See appendix. \end{proof} 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. @@ -306,6 +309,28 @@ In some applications, such as in quantum mechanics, it is more common to see it \end{align*} \end{proof} +Now, using the solutions to the Legendre equation we can construct the solution to the more general problem: +\begin{align} + \left( 1 - x^2 \right) \frac{d^2 y}{dx^2} &- 2x \frac{dy}{dx} \nonumber \\ + & + \left[ n(n+1) - \frac{m^2}{1 - x^2} \right] y(x) = 0. + \label{eqn:legendre-as} +\end{align} + +This equation is considerably more difficult, and again, we will just analyze the solution. + +\begin{definition}[Associated Legendre Polynomials] + Let \(m \in \mathbb{N}_0\). The polynomials + \begin{equation} \label{eqn:legendre-poly-as} + P_{m, n} (x) = \left( 1 - x^2 \right)^{m/2} \frac{d^m}{dx^m} P_n (x), + \end{equation} + are called the associated Legendre polynomials. +\end{definition} + +\begin{lemma} + The associated Legendre polynomials \eqref{eqn:legendre-poly-as} are solutions to the associated Legendre differential equation \eqref{eqn:legendre-as}. +\end{lemma} +\begin{proof} See appendix. \end{proof} + \subsection{Spherical harmonics} \clearpage @@ -374,6 +399,7 @@ from which we can extract the recurrence relation \end{gather*} % TODO: finish reviewing proof + \texttt{[TODO: finish copying proof]} \if 0 We can derive a recursion formula for $a_{k+2}$ from Eq.\eqref{eq:condition_2}, which can be expressed as diff --git a/notes/build/FourierOnS2.pdf b/notes/build/FourierOnS2.pdf Binary files differindex d40262c..d60d9bf 100644 --- a/notes/build/FourierOnS2.pdf +++ b/notes/build/FourierOnS2.pdf |