diff options
| -rw-r--r-- | buch/papers/kugel/references.bib | 16 | ||||
| -rw-r--r-- | buch/papers/kugel/spherical-harmonics.tex | 21 |
2 files changed, 31 insertions, 6 deletions
diff --git a/buch/papers/kugel/references.bib b/buch/papers/kugel/references.bib index e13b67c..7603f2f 100644 --- a/buch/papers/kugel/references.bib +++ b/buch/papers/kugel/references.bib @@ -275,4 +275,20 @@ This photo was taken during studies that resulted in the publication: Hope, C, S urldate = {2022-08-27}, date = {2022-04-27}, doi = {10.5281/ZENODO.6497293}, +} + +@article{davari_new_2013, + title = {New Implementation of Legendre Polynomials for Solving Partial Differential Equations}, + volume = {04}, + issn = {2152-7385, 2152-7393}, + url = {http://www.scirp.org/journal/doi.aspx?DOI=10.4236/am.2013.412224}, + doi = {10.4236/am.2013.412224}, + pages = {1647--1650}, + number = {12}, + journaltitle = {Applied Mathematics}, + shortjournal = {{AM}}, + author = {Davari, Ali and Ahmadi, Abozar}, + urldate = {2022-08-28}, + date = {2013}, + file = {Full Text:/Users/npross/Zotero/storage/A8XM56WK/Davari and Ahmadi - 2013 - New Implementation of Legendre Polynomials for Sol.pdf:application/pdf}, }
\ No newline at end of file diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex index 02d93e9..0fb6557 100644 --- a/buch/papers/kugel/spherical-harmonics.tex +++ b/buch/papers/kugel/spherical-harmonics.tex @@ -190,7 +190,6 @@ require a dedicated section of its own. \begin{figure} \centering - % \kugelplaceholderfig{.8\textwidth}{5cm} \includegraphics[ scale = 1.2, trim = 0 40 0 0, clip, @@ -327,7 +326,14 @@ obtain the \emph{associated Legendre functions}. \end{equation} are known as Ferrers or associated Legendre functions. \end{definition} -The constraint $|m|<n$, can be justified by considering eq.\eqref{kugel:eq:associated_leg_func}, where we differentiate $m+n$ times. We all know that a differentiation, to be well defined, must have an order that is greater than zero \kugeltodo{is that always true?}. Furthermore, it can be seen that this derivative is applied on a polynomial of degree $2n$. As is known from Calculus 1, if you derive a polynomial of degree $2n$ more than $2n$ times, you get zero, that would be a trivial solution. This is the power of zero: It is almost always a (boring) solution. +The constraint $|m|<n$, can be justified by considering equation +\eqref{kugel:eq:associated_leg_func}, where we differentiate $m+n$ times. We all +know that a differentiation, to be well defined, must have an order that is +greater than zero. Furthermore, it can be seen that this derivative is applied +on a polynomial of degree $2n$. As is known from Calculus 1, if you derive a +polynomial of degree $2n$ more than $2n$ times, you get zero, that would be a +trivial solution. This is the power of zero: It is almost always a (boring) +solution. We can thus summarize these two conditions by writing: \begin{equation*} @@ -344,8 +350,8 @@ section \ref{kugel:sec:construction:eigenvalue}. We had left off in the middle of the separation, were we had used the Ansatz $f(\vartheta, \varphi) = \Theta(\vartheta) \Phi(\varphi)$ to find that $\Phi(\varphi) = e^{im\varphi}$, and we were solving for $\Theta(\vartheta)$. As you may recall, previously we -performed the substitution $z = \cos \vartheta$. Now we can finally bring back the -solution to the associated Legendre equation $P^m_n(z)$ into the $\vartheta$ +performed the substitution $z = \cos \vartheta$. Now we can finally bring back +the solution to the associated Legendre equation $P^m_n(z)$ into the $\vartheta$ domain and combine it with $\Phi(\varphi)$ to get the full result: \begin{equation*} f(\vartheta, \varphi) @@ -709,8 +715,11 @@ To start this journey, we can first write the following equations, which relate \end{subequations} Much of the effort will be proving this bunch of equalities. Then, in the second part, where we will derive the recursion equations for $Y^m_n(\vartheta,\varphi)$, we will basically reuse the ones presented above. -Maybe it is worth mentioning at least one use case for these relations: In some software implementations (that include lighting computations in computer graphics, antenna modelling softwares, 3-D modelling in medical applications, etc.) -they are widely used, as they lead to better numerical accuracy and computational cost lower by a factor of six\cite{usecase_recursion_paper}. +Maybe it is worth mentioning at least one use case for these relations: In some +software implementations (that include lighting computations in computer +graphics, antenna modelling software, 3-D modelling in medical applications, +etc.) they are widely used, as they lead to better numerical accuracy and +computational cost lower by a factor of six \cite{davari_new_2013}. \begin{enumerate}[(i)] \item \begin{proof} |