How can I make this table prettier?

Question

The following code gives the following result:

\documentclass{article}
\usepackage{longtable,amsmath}
\begin{document}
\begin{longtable}{|p{0.05\textwidth}|p{0.47\textwidth}|p{0.47\textwidth}|}
    \hline
    &\multicolumn{1}{|c|}{$\Delta v=0$}&\multicolumn{1}{c|}{$-\Delta v=\lambda v$}\\
    \hline
    2D & \begin{multline*}C+D\ln r \\
    +\left(A_nr^n+ \frac{B_n}{r^n}\right)(C_n\cos n\theta+D_n\cos n\theta) \end{multline*} & \begin{multline*} (A_nJ_n(\sqrt\lambda r)+B_nN_n(\sqrt\lambda r)) \\
    \times(C_n\cos n\theta+D_n\sin n\theta)\end{multline*} \\
    \hline
    3D & \begin{multline*} \left(A_lr^l+\frac{B_l}{r^{l+1}}\right)P^m_l(\cos\theta) e^{im\phi}\end{multline*} & \begin{multline*}\small \frac{A_lJ_{l+1/2}(\sqrt\lambda r)+B_lN_{l+1/2}(\sqrt\lambda r)}{\sqrt r} \\
    \times P^l_m(\cos\theta)e^{im\phi} \end{multline*}
    \hline
    \end{longtable}
\end{document}

How can I make this table prettier? Especially, (1) How can I eliminate the "double vertical line" on the left of the $\Delta v=0$? (2) How can I make "2D" and "3D" in the center of the table?

Answer 1

To remove the double vertical line in the header row, use {c|} instead of {|c|} and to vertically center the dontents in the first row, replace the p by m type columns:

\documentclass{article}
\usepackage{geometry}
\usepackage{longtable,amsmath}
\usepackage{array}
\begin{document}

\begin{longtable}{|l|m{0.47\textwidth}|m{0.47\textwidth}|}
    \hline
    &\multicolumn{1}{c|}{$\Delta v=0$}&\multicolumn{1}{c|}{$-\Delta v=\lambda v$}\\
    \hline
    2D & \begin{multline*}C+D\ln r \\
    +\left(A_nr^n+ \frac{B_n}{r^n}\right)(C_n\cos n\theta+D_n\cos n\theta) \end{multline*} & \begin{multline*} (A_nJ_n(\sqrt\lambda r)+B_nN_n(\sqrt\lambda r)) \\
    \times(C_n\cos n\theta+D_n\sin n\theta)\end{multline*} \\
    \hline
    3D & \begin{multline*} \left(A_lr^l+\frac{B_l}{r^{l+1}}\right)P^m_l(\cos\theta) e^{im\phi}\end{multline*} & \begin{multline*}\small \frac{A_lJ_{l+1/2}(\sqrt\lambda r)+B_lN_{l+1/2}(\sqrt\lambda r)}{\sqrt r} \\
    \times P^l_m(\cos\theta)e^{im\phi} \end{multline*} \\
    \hline
    \end{longtable}
\end{document}

Answer 2

There is an article that nicely discusses how to make attractive tables using LaTeX, “Tables in LATEX2e: Packages and Methods.” Drawing on the advice given in that article, I would recommend removing all of your vertical rules and using the commands \toprule, \midrule, and \bottomrule (from the booktabs package) to distinguish the top and bottom rules.