From cba8ece37bbd9d57322f76f4b6d1584464dfb7b1 Mon Sep 17 00:00:00 2001 From: Karl Berry Date: Wed, 16 Feb 2022 22:43:49 +0000 Subject: asmeconf (15feb22) git-svn-id: svn://tug.org/texlive/trunk@62047 c570f23f-e606-0410-a88d-b1316a301751 --- .../asmeconf-template-widetext.pdf | Bin 0 -> 2931161 bytes .../asmeconf-template-widetext.tex | 348 +++++++++++++++++++++ .../asmeconf-wide-equation-example/asmewide.sty | 128 ++++++++ .../tesseral-harmonic.pdf | Bin 0 -> 1237557 bytes .../zonal-harmonic2.pdf | Bin 0 -> 1221215 bytes 5 files changed, 476 insertions(+) create mode 100644 Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.pdf create mode 100644 Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.tex create mode 100644 Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmewide.sty create mode 100644 Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/tesseral-harmonic.pdf create mode 100644 Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/zonal-harmonic2.pdf (limited to 'Master/texmf-dist/doc/latex/asmeconf/examples') diff --git a/Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.pdf b/Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.pdf new file mode 100644 index 00000000000..3b915adbcc0 Binary files /dev/null and b/Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.pdf differ diff --git a/Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.tex b/Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.tex new file mode 100644 index 00000000000..209cf294ca1 --- /dev/null +++ b/Master/texmf-dist/doc/latex/asmeconf/examples/asmeconf-wide-equation-example/asmeconf-template-widetext.tex @@ -0,0 +1,348 @@ +%% +%% Examples of a widetext environment for setting wide equations in the asmeconf class. +%% Copyright (c) 2021 John H. Lienhard. Use under the MIT license: https://ctan.org/license/mit +%% +%% USAGE: * \begin{widetext} ...wide material here... \end{widetext} +%% OPTIONAL ARGUMENTS: +%% * \begin{widetext[N] .. changes upper/lower separation of wide material from default 5pt to Npt +%% * \begin{widetext}[][tbn]: t = top line only; b = bottom line only; n = no lines. BOTH arguments are REQUIRED, even if first is left empty. +%% +%% The widetext environment can only appear once per page. It clashes with floats and footnotes, as discussed herein. +%% +%% NB: the strip environment from cuted is incompatible with the [lineno] option to asmeconf! + +\documentclass[nofoot,colorlinks,balance,pdf-a]{asmeconf} + +\def\ACwidetextversion{1.0} +\def\ACwidetextdate{January 10, 2022} + +\usepackage{lipsum}% Latin filler text +\usepackage{asmewide} + + +\begin{filecontents}{asme-wide-equations.bib} +@online{lienhard2021, + author = {Lienhard, John H., V}, + title = {{\texttt{asmeconf}}: A template for {ASME} conference papers}, + organization = {Comprehensive \TeX\ Archive Network}, + version = {{\versionno}}, + year = {2021}, + url = {https://ctan.org/pkg/asmeconf}, + urldate = {{\today}}, +} +@online{tolucsis1, + author = {Sigitas Tolu\v{s}is}, + year = {2021}, + title = {The \texttt{cuted} package}, + version = {2.0}, + organization = {Comprehensive \TeX\ Archive Network}, + url = {https://ctan.org/pkg/cuted}, + urldate = {Dec. 30, 2021}, +} +@online{tolucsis2, + author = {Sigitas Tolu\v{s}is}, + year = {2021}, + title = {The \texttt{sttools} collection}, + version = {3.0}, + organization = {Comprehensive \TeX\ Archive Network}, + url = {https://ctan.org/pkg/sttools}, + urldate = {Dec. 30, 2021}, +} +@book{stakgold, +author = {Ivar Stakgold}, +title = {Boundary Value Problems of Mathematical Physics}, +year = {1967}, +publisher = {Macmillan}, +address = {New York}, +} +\end{filecontents} + +\hypersetup{% + pdfauthor={John H. Lienhard}, + pdftitle={Wide Equations in asmeconf.cls}, + pdfkeywords={ASME conference paper, LaTeX template, wide equations, widetext}, + pdfsubject = {Examples of setting wide equations in the asmeconf LaTeX template}, + pdfurl={https://ctan.org/pkg/asmeconf}, + pdflicenseurl={https://ctan.org/pkg/asmeconf}, +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\ConfName{Proceedings of the \texttt{asmeconf}\linebreak International Examples Congress and Exposition} +\ConfAcronym{AIECE22} +\ConfDate{January 10, 2022} +\ConfCity{Cambridge, MA} +\PaperNo{AIECE2022-0003} + +\SetAuthors{John H.\ Lienhard V\affil{}\CorrespondingAuthor{lienhard@mit.edu}} +\SetAffiliation{}{Fellow of ASME \\ +Rohsenow Kendall Heat Transfer Laboratory, \\ +Department of Mechanical Engineering,\\ +Massachusetts Institute of Technology, \\ +Cambridge, MA 02139 USA +} + +\title{Wide Equations in asmeconf.cls} +\maketitle + +\versionfootnote{Examples of \texttt{widetext} in \texttt{asmeconf}. Version \ACwidetextversion, \ACwidetextdate} + +\keywords{ASME conference paper, \LaTeX\ template, wide equations, asmeconf} + +\begin{abstract} +This paper gives several examples of typesetting very wide equations with {\upshape\LaTeX} in the {\upshape\texttt{asmeconf}} class~{\upshape\cite{lienhard2021}} using {\upshape\texttt{asmewide.sty}}. +The style defines a the {\upshape\texttt{widetext}} environment, built on the 2021 release of {\upshape\texttt{cuted.sty}~\cite{tolucsis1}} from the +{\upshape\texttt{sttools}} bundle~{\upshape\cite{tolucsis2}}, which is available from CTAN, \hrefurl{http://ctan.org}{ctan.org}. +Significant hand-fitting around pagebreaks, floats, and footnotes is required to obtain good results. Users can change the source file to explore the behavior and limitations of the {\upshape\texttt{widetext}} environment. \textcolor{red}{Only the text in \textbf{red} in this document meant +to be read---the rest is simply filler to aid in layout.} +\end{abstract} + + +\section{Introduction} +\lipsum[1-2] +\section{Section} +\lipsum[3] + +%%%%%%%%%%%%%%%%% begin two column figure %%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{figure*}[t] +\begin{subfigure}[c]{0.495\textwidth} +\centering{\includegraphics{zonal-harmonic2.pdf}}% +\subcaption{\label{fig:zonal}} +\end{subfigure} +%%%%%%%% don't leave a break here +\begin{subfigure}[c]{0.495\textwidth} +\centering{\includegraphics{tesseral-harmonic.pdf}}% +\subcaption{\label{fig:tesseral}}% +\end{subfigure}% +\caption{A figure with two subfigures: (a) Zonal harmonic $n=1, m=0$, (b) Tesseral harmonic $n=2, m=3$. See Appendix~\ref{sec:sph-har}.\label{fig:1}} +\end{figure*} +%%%%%%%%%%%%%%%%%%% end two column figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\lipsum[4] +\subsection{Subsection} +\lipsum[5-6] + + +%%%%%%%%%%%%%%%%%%%% Example WT1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Single Wide Equation on the Page} + +\textcolor{red}{Equation~\eqref{eqn:WT1} is an equation with a matrix that is too large to fit into one column. A multiline math environment will not help because the equation cannot be broken into parts that each fit into a column. The \texttt{widetext} environment solves the problem.} + +\textcolor{red}{A two-column wide figure, Fig.~\ref{fig:1}, has floated from a previous page to the top of this page, but does not interfere with the \texttt{widetext} material (a one column float would interfere).} + +\begin{widetext} +\begin{equation}\label{eqn:WT1} +\mathbf{WT1:}\quad +\mathfrak{W}(\bm{\Phi})= \begin{Vmatrix} +\dfrac\varphi{(\varphi_1,\varepsilon_1)} & 0 & \hdotsfor{4} & 0 & \\[\jot] +\dfrac{\varphi k_{21}}{(\varphi_2,\varepsilon_1)} & \dfrac\varphi{(\varphi_2,\varepsilon_2)} & 0 & \hdotsfor{3} & 0 \\[\jot] +\dfrac{\varphi k_{31}}{(\varphi_3,\varepsilon_1)} &\dfrac{\varphi k_{32}}{(\varphi_3,\varepsilon_2)} & \dfrac\varphi{(\varphi_3,\varepsilon_3)}& 0 & \hdotsfor{2} & 0 \\[\jot] +\vdots & & & \smash{\rotatebox{15}{$\ddots$}} & & & \vdots \\[\jot] +\dfrac{\varphi k_{n-2\, 1}}{(\varphi_{n-2},\varepsilon_1)} & +\dfrac{\varphi k_{n-2\, 2}}{(\varphi_{n-2},\varepsilon_2)} &\hdotsfor{1} & \dfrac{\varphi k_{n-2\,n-3}}{(\varphi_{n-2},\varepsilon_{n-3})} & \dfrac\varphi{(\varphi_{n-2},\varepsilon_{n-2})}& 0& 0 \\[\jot] +\dfrac{\varphi k_{n-1\, 1}}{(\varphi_{n-1},\varepsilon_1)} & \dfrac{\varphi k_{n-1\, 2}}{(\varphi_{n-1},\varepsilon_2)} &\hdotsfor{2} & +\dfrac{\varphi k_{n-1\,n-2}}{(\varphi_{n-1},\varepsilon_{n-2})}& \dfrac{\varphi}{(\varphi_{n-1},\varepsilon_{n-1})} & 0 \\[\jot] +\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)} & \dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)} & \hdotsfor{3} & +\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})} & \dfrac{\varphi}{(\varphi_n,\varepsilon_n)} +\end{Vmatrix} +\end{equation} +\end{widetext} + +\lipsum[7-8] + + +%%%%%%%%%%%%%%%%%%%% Examples WT2 & WT3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Two Wide Equations on the Page} + +\lipsum[9-10] + +\begin{widetext} +\begin{equation}\mathbf{WT2:} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy + \ne \frac{1}{\sqrt{\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy}} +\end{equation} + +\textcolor{red}{In this case, we have a pair of wide equations on the same page. The \texttt{widetext} environment cannot be used twice on the same page! To resolve the conflict, we remain in single column mode between the two equations.} + +\textcolor{red}{This page also includes a single column float, Table~\ref{tab:2}. This float must come after the \texttt{widetext} environment. We use the \texttt{\textbackslash begin\{table\}[b]} option to force the table to the bottom of the column. The two column table, Table~\ref{tab:4}, floats to the top of the next page and creates no problems.} + +\begin{equation}\mathbf{WT3:} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy + \ne \frac{1}{\sqrt{\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy}} +\end{equation} +\end{widetext} + + +%%%%%%%%%%%%%%% begin single column table %%%%%%%%%%%%%%%%%%%%%% +\begin{table}[b] +\caption{Table with more complicated columns}\label{tab:2}% +\centering{% +\begin{tabular}{!{\hspace*{0.5cm}} >{\raggedright\hangindent=1em} p{3cm} d{3} @{\hspace*{1cm}} d{3} !{\hspace*{0.5cm}}} +\hline\hline +\rule{0pt}{10pt} Experiment & \multicolumn{1}{c@{\hspace*{1cm}}}{$u$ [m/s]} & \multicolumn{1}{c!{\hspace*{0.5cm}}}{$T$ [\textdegree C]} \\[1pt] +\hline +The first experiment we ran this morning & 124.3 & 68.3 \rule{0pt}{10pt} \\ +The second experiment we ran this morning & 82.50 & 103.46 \\ +Our competitor's data & 72.321 & 141.384 \\[1pt] +\hline\hline +\end{tabular} +} +\end{table} +%%%%%%%%%%%%%%%% end table %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%% begin two column table %%%%%%%%%%%%%%%%%%%%%%%%% +\begin{table*}[t] +\caption{A table spanning two columns}\label{tab:4}% +\centering{% +\begin{tabular*}{0.8\textwidth}{@{\hspace*{1.5em}}@{\extracolsep{\fill}}ccc!{\hspace*{3.em}}ccc@{\hspace*{1.5em}}} +\hline\hline +\multicolumn{1}{@{\hspace*{1.5em}}c}{$x$\rule{0pt}{11pt}} & +\multicolumn{1}{c}{$\textrm{erf}(x)$} & +\multicolumn{1}{c!{\hspace*{3.em}}}{$\textrm{erfc}(x)$} & +\multicolumn{1}{c}{$x$} & +\multicolumn{1}{c}{$\textrm{erf}(x)$} & +\multicolumn{1}{c@{\hspace*{1.5em}}}{$\textrm{erfc}(x)$} \\ \hline +0.00 & 0.00000 & 1.00000 & 1.10 & 0.88021 & 0.11980\rule{0pt}{11pt} \\ +0.05 & 0.05637 & 0.94363 & 1.20 & 0.91031 & 0.08969 \\ +0.10 & 0.11246 & 0.88754 & 1.30 & 0.93401 & 0.06599 \\ +0.15 & 0.16800 & 0.83200 & 1.40 & 0.95229 & 0.04771 \\ +0.20 & 0.22270 & 0.77730 & 1.50 & 0.96611 & 0.03389 \\ +0.30 & 0.32863 & 0.67137 & 1.60 & 0.97635 & 0.02365 \\ +0.40 & 0.42839 & 0.57161 & 1.70 & 0.98379 & 0.01621 \\ +0.50 & 0.52050 & 0.47950 & 1.80 & 0.98909 & 0.01091 \\ +0.60 & 0.60386 & 0.39614 & 1.82\makebox[0pt][l]{14} & 0.99000 & 0.01000 \\ +0.70 & 0.67780 & 0.32220 & 1.90 & 0.99279 & 0.00721 \\ +0.80 & 0.74210 & 0.25790 & 2.00 & 0.99532 & 0.00468 \\ +0.90 & 0.79691 & 0.20309 & 2.50 & 0.99959 & 0.00041 \\ +1.00 & 0.84270 & 0.15730 & 3.00 & 0.99998 & 0.00002 \\[2pt] +\hline\hline +\end{tabular*} +} +\end{table*} +%%%%%%%%%%%%%%%% end table %%%%%%%%%%%%%%%%%%% + +\lipsum[17-24] + + +%%%%%%%%%%%%%%%%%%%% Examples WT4 & WT5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Wide Equation Pair Split Across Page Break and Followed by Wide Equation} + +\textcolor{red}{Here, we turn off automatic top line so that it does not appear again at the top of the next page: \texttt{\textbackslash TopLineOff}. We instead issue a command to create the top line inside the \texttt{widetext} environment, \texttt{\textbackslash LineAbove}. We must remember to restore the automatic top line after we exit the \texttt{widetext} environment: \texttt{\textbackslash TopLineOn}.} + +\textcolor{red}{The \texttt{\textbackslash newpage} command may optionally be used between the equations to force the second one onto the following page, e.g., try removing the source code line \texttt{\textbackslash lipsum[27]} with and without \texttt{\textbackslash newpage}.} + +\begin{widetext} +\begin{equation}\mathbf{WT4:} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy + \ne \frac{1}{\sqrt{\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy}} +\end{equation}%\newpage +\begin{equation}\mathbf{WT5:} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy + \ne \frac{1}{\sqrt{\int_a^b\biggl\{g(y)^2\int_a^bf^2+f(y)^2 + \int_a^b g^2-2f(y)g(y)\int_a^b fg\biggr\}\,dy}} +\end{equation} + +\textcolor{red}{In this case, we again have a pair of wide equations on the same page, so we stay in single column mode +until both are done\footnotemark. The single column table, Table~\ref{tab:3}, is forced to the bottom of the page using the \texttt{[b]} option.} + +\lipsum[32-33] + +\begin{equation}\mathbf{WT6:} +\int_a^b\biggl\{\int_a^b[f(x)^2g(y)^2+f(y)^2g(x)^2] + -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy + \ne \frac{1}{\sqrt{\int_a^b\biggl\{g^2\int_a^bf^2+f^2 + \int_a^b g^2-2fg\int_a^b fg\biggr\}\,dy}} +\end{equation} +\end{widetext} +\footnotetext{\textcolor{red}{The code from \texttt{cuted.sty} doesn't play well with footnotes, so we issue a \texttt{\textbackslash footnotemark} command inside the wide material and a separate \texttt{\textbackslash footnotetext\{..\}} command outside the wide environment.}}% +\lipsum[34-37] + +%%%%%%%%%%%%%%%%%%% begin linewidth table %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{table}[b] +\newcolumntype{C}{>{$}c<{$}} % math-mode version of "c" column type, from array package +\caption{\label{tab:3}Table at full column width with columns in math mode} +\centering{% +\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}CCCC@{\extracolsep{\fill}}} +\hline\hline +X_{z} & X_{c} & X_{c,m} & X_{c,2}\rule{0pt}{11pt}\\ + 3.92069 & 5.70943 & 6.32429 & 7.08757\\[2pt] +\varepsilon (T_1) & \varepsilon^i (T_1) & \varepsilon^i (T_m) & \alpha (T_1, T_2)\\ +0.7258 & 0.6237 & 0.6807 & 0.7964 \\[2pt] +q_\textrm{gray} & q_\textrm{int, $T_1$} & q_\textrm{int, $T_m$} & q_\textrm{exact}\\ +400.2 & 462.1 & 371.0 & 371.8 \\[1pt] +\hline\hline +\end{tabular*} +} +\end{table} +%%%%%%%%%%%%%%%%%%%% end linewidth table %%%%%%%%%%%%%%%%%%%%%%% + +\lipsum[40-48] + +\textcolor{red}{For eqn.~\eqref{eqn:WT7}, we drop the bottom line, keeping the top line and increasing the vertical space a bit: \texttt{\textbackslash begin\{widetext\}[8][t]}.} + +\begin{widetext}[8][t] +\begin{equation}\label{eqn:WT7} +\mathbf{WT7:}\quad +\cfrac{1}{1+ \cfrac{1}{abcxyz+(ax^2-by^3+cz^4)(\alpha\chi^2-\beta\upsilon^3+\kappa\zeta^4)(ax^4-by^3+cz^2)(a^2x^2-by^3+c^2z^2)}} +\end{equation} +\end{widetext} + +\lipsum[50-60] + +\textcolor{red}{In this case, we drop the top line: \texttt{\textbackslash begin\{widetext\}[][b]}.} + +\vskip 80pt% <== an extra skip to push this widetext over the edge + +\begin{widetext}[][b] +\begin{equation}\label{eqn:WT8} +\mathbf{WT8:}\quad +\mathfrak{W}(\bm{\Phi})= \begin{Vmatrix} +\dfrac\varphi{(\varphi_1,\varepsilon_1)} & 0 & \hdotsfor{4} & 0 & \\[\jot] +\dfrac{\varphi k_{21}}{(\varphi_2,\varepsilon_1)} & \dfrac\varphi{(\varphi_2,\varepsilon_2)} & 0 & \hdotsfor{3} & 0 \\[\jot] +\dfrac{\varphi k_{31}}{(\varphi_3,\varepsilon_1)} &\dfrac{\varphi k_{32}}{(\varphi_3,\varepsilon_2)} & \dfrac\varphi{(\varphi_3,\varepsilon_3)}& 0 & \hdotsfor{2} & 0 \\[\jot] +\vdots & & & \smash{\rotatebox{15}{$\ddots$}} & & & \vdots \\[\jot] +\dfrac{\varphi k_{n-2\, 1}}{(\varphi_{n-2},\varepsilon_1)} & +\dfrac{\varphi k_{n-2\, 2}}{(\varphi_{n-2},\varepsilon_2)} &\hdotsfor{1} & \dfrac{\varphi k_{n-2\,n-3}}{(\varphi_{n-2},\varepsilon_{n-3})} & \dfrac\varphi{(\varphi_{n-2},\varepsilon_{n-2})}& 0& 0 \\[\jot] +\dfrac{\varphi k_{n-1\, 1}}{(\varphi_{n-1},\varepsilon_1)} & \dfrac{\varphi k_{n-1\, 2}}{(\varphi_{n-1},\varepsilon_2)} &\hdotsfor{2} & +\dfrac{\varphi k_{n-1\,n-2}}{(\varphi_{n-1},\varepsilon_{n-2})}& \dfrac{\varphi}{(\varphi_{n-1},\varepsilon_{n-1})} & 0 \\[\jot] +\dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)} & \dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)} & \hdotsfor{3} & +\dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})} & \dfrac{\varphi}{(\varphi_n,\varepsilon_n)} +\end{Vmatrix} +\end{equation} +\end{widetext} + +\lipsum[55-56] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\appendix +\section{Spherical harmonics\label{sec:sph-har}} + +Without getting into the details, a regular function $f(\theta,\phi)$ on the surface of the unit sphere may be written +\begin{equation} +f(\theta,\phi) = \sum_{n=0}^\infty \sum_{m=-n}^n f_{m,n} Y_n^m(\theta,\phi) +\end{equation} +for $Y_n^m(\theta,\phi) = e^{i m\phi}P^{|m|}_n(\cos\theta)$, for $|m|