Cannot split long subequations

Question

I do not seem to be able to split two long sub-equations. I've tried multline, align, split and a few other possibilities but none of them have worked; I reckon a main issue is that I've got brackets everywhere (very long equations indeed!).

What I'd like is to have only two numbered equations (1a and 1b) but for these equations to fit within a standard page width. Any help would be greatly appreciated.

This is my working code, but I'm not sure where to go from here:

\begin{subequations}\label{eqn:MagneticField3D}
    \begin{gather}
        B_{x}(r,x)=\frac{N I \mu_{0}}{2\pi}\left(\frac{1}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}\left(\mathcal{K}(k(r,x-d/2)^{2})+\frac{a^{2}-r^{2}-(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)-\frac{1}{\sqrt{(a+r)^{2}+(x+d/2)^{2}}}\left(\mathcal{K}(k(r,x+d/2)^{2})+\frac{a^{2}-r^{2}-(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)\right), \\
        B_{r}(r,x)=\frac{N I \mu_{0}}{2\pi r}\left(\frac{x-d/2}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}\left(-\mathcal{K}(k(r,x-d/2)^{2})+\frac{a^{2}+r^{2}+(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)-\frac{x+d/2}{\sqrt{(a+r)^{2}+(x+d/2)^{2}}}\left(-\mathcal{K}(k(r,x+d/2)^{2})+\frac{a^{2}+r^{2}+(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)\right).
    \end{gather}
\end{subequations}

Answer 1

Here is a solution that does not require introducing new notation. It makes use of align, plus \left. and \right. where necessary, as well as \vphantom to get the correct sizing of the brackets and parentheses. I also changed the outer parentheses to square brackets because I think it's easier to read, but obviously that's a stylistic choice.

\documentclass{article}

\usepackage{amssymb,amsmath}

\begin{document}

\begin{subequations}\label{eqn:MagneticField3D}

    \begin{align}
        B_{x}(r,x) = & \frac{N I \mu_{0}}{2\pi}\left[\frac{1}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}\left(\mathcal{K}(k(r,x-d/2)^{2}) 
        \vphantom{\frac{a^{2}-r^{2}-(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}} % to size the ( correctly
        \right.\right. \nonumber \\
        & \left.\left. +\frac{a^{2}-r^{2}-(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right) \right. \nonumber \\
       & \left. -\frac{1}{\sqrt{(a+r)^{2}+(x+d/2)^{2}}}\left(\mathcal{K}(k(r,x+d/2)^{2}) 
       \vphantom{\frac{a^{2}-r^{2}-(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}} % to size the ( correctly
       \right.\right. \nonumber \\
       &  \left.\left. +\frac{a^{2}-r^{2}-(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)
       \vphantom{\frac{1}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}} % to size the ] correctly
       \right], \\
        B_{r}(r,x) = & \frac{N I \mu_{0}}{2\pi r}\left[\frac{x-d/2}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}\left(-\mathcal{K}(k(r,x-d/2)^{2}) 
        \vphantom{\frac{a^{2}+r^{2}+(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}} % to size the ( correctly
        \right.\right. \nonumber \\
            & \left.\left. +\frac{a^{2}+r^{2}+(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right) \right. \nonumber \\
    & \left. -\frac{x+d/2}{\sqrt{(a+r)^{2}+(x+d/2)^{2}}}\left(-\mathcal{K}(k(r,x+d/2)^{2}) 
    \vphantom{\frac{a^{2}+r^{2}+(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}} % to size the ( correctly
    \right.\right. \nonumber \\
    & \left.\left. +\frac{a^{2}+r^{2}+(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)
    \vphantom{\frac{x-d/2}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}} % to size the ] correctly
    \right].
    \end{align}
\end{subequations}

\end{document}

You could also add some \hspace to bump out the second lines of the "inner" parentheses:

\documentclass{article}

\usepackage{amssymb,amsmath}

\begin{document}

\begin{subequations}\label{eqn:MagneticField3D}

    \begin{align}
        B_{x}(r,x) = & \frac{N I \mu_{0}}{2\pi}\left[\frac{1}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}\left(\mathcal{K}(k(r,x-d/2)^{2}) 
        \vphantom{\frac{a^{2}-r^{2}-(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}} % to size the ( correctly
        \right.\right. \nonumber \\
        & \hspace{1cm} \left.\left. +\frac{a^{2}-r^{2}-(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right) \right. \nonumber \\
       & \left. -\frac{1}{\sqrt{(a+r)^{2}+(x+d/2)^{2}}}\left(\mathcal{K}(k(r,x+d/2)^{2}) 
       \vphantom{\frac{a^{2}-r^{2}-(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}} % to size the ( correctly
       \right.\right. \nonumber \\
       &  \hspace{1cm} \left.\left. +\frac{a^{2}-r^{2}-(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)
       \vphantom{\frac{1}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}} % to size the ] correctly
       \right], \\
        B_{r}(r,x) = & \frac{N I \mu_{0}}{2\pi r}\left[\frac{x-d/2}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}\left(-\mathcal{K}(k(r,x-d/2)^{2}) 
        \vphantom{\frac{a^{2}+r^{2}+(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}} % to size the ( correctly
        \right.\right. \nonumber \\
            & \hspace{1cm} \left.\left. +\frac{a^{2}+r^{2}+(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right) \right. \nonumber \\
    & \left. -\frac{x+d/2}{\sqrt{(a+r)^{2}+(x+d/2)^{2}}}\left(-\mathcal{K}(k(r,x+d/2)^{2}) 
    \vphantom{\frac{a^{2}+r^{2}+(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}} % to size the ( correctly
    \right.\right. \nonumber \\
    & \hspace{1cm} \left.\left. +\frac{a^{2}+r^{2}+(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}\mathcal{E}(k(r,x-d/2)^{2})\right)
    \vphantom{\frac{x-d/2}{\sqrt{(a+r)^{2}+(x-d/2)^{2}}}} % to size the ] correctly
    \right].
    \end{align}
\end{subequations}

\end{document}

Answer 2

You can probably reduce it down to. Dues to the width of the remaining fractions, it did not seem forth it to reduce these.

\documentclass[a4paper]{article}
\usepackage{amsmath}
\begin{document}

To reduce complexity, define
\begin{align*}
  A &= \frac{N I \mu_0}{2\pi}, \\
  B &= \sqrt{(a+r)^{2}+(x-d/2)^{2}},\\
  C &= \mathcal{K}(k(r,x-d/2)^{2}), \\
  D &= \mathcal{E}(k(r,x-d/2)^{2}), \\
  E  &= \sqrt{(a+r)^{2}+(x+d/2)^{2}}, \\
  F &= \mathcal{K}(k(r,x+d/2)^{2}),
\end{align*}
\begin{subequations}\label{eqn:MagneticField3D}
  \begin{gather}
    \begin{aligned}[b]
      B_{x}(r,x)=A\Biggl(&\frac{1}{B}\Biggl(C+\frac{a^{2}-r^{2}-(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}D\Biggr)
      \\
      &-\frac{1}{E}\Biggl(F+\frac{a^{2}-r^{2}-(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}D\Biggr)\Biggr),
    \end{aligned}
    \\
    \begin{aligned}[b]
      B_{r}(r,x)=
      \frac{A}{r}\Biggl(&\frac{x-d/2}{B}\Biggl(-C+\frac{a^{2}+r^{2}+(x-d/2)^{2}}{(a-r)^{2}+(x-d/2)^{2}}D\Biggr)
      \\
      &-\frac{x+d/2}{E}\Biggl(-F+\frac{a^{2}+r^{2}+(x+d/2)^{2}}{(a-r)^{2}+(x+d/2)^{2}}D\Biggr)\Biggr).
    \end{aligned}
  \end{gather}
\end{subequations}

\end{document}

Answer 3

Another possibility, with geometry (for more sensible margins) and nccmath (for medium-sized formulæ, ~80 % of \displaystyle, and the fleqn environment):

\documentclass[a4paper]{article}
\usepackage[ showframe]{geometry}
\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{mathtools, nccmath}

\begin{document}

\begin{subequations}\label{eqn:MagneticField3D}
\begin{fleqn}
    \begin{gather}
\raisetag{11.5ex}
\begin{aligned}
B_{x}(r,x)= \medmath{\frac{N I \mu_{0}}{2\pi} \! \left(\frac{1}{\sqrt{(a+r)^{2}+\bigl(x-\mfrac{d}{2}\bigr)^{2}}}\Biggl(\mathcal{K}\Bigl(k\bigl(r,x-\mfrac{d}{2}\bigr)^{2}\Bigr) + \frac{a^{2}-r^{2}-\bigl(x-\mfrac{d}{2}\bigr)^{2}}{(a-r)^{2}+\bigl(x-\mfrac{d}{2}\bigr)^{2}} \mathcal{E}\Bigl(k\bigl(r,x-\mfrac{d}{2}\bigr)^{2}\Bigr)\Biggr) \right.} \\
  \medmath{-\left. \frac{1}{\sqrt{(a+r)^{2}+\bigl(x + \mfrac{d}{2}\bigr)^{2}}}\Biggl(\mathcal{K}\Bigl(k\bigl(r,x + \mfrac{d}{2}\bigr)^{2}\Bigr) + \frac{a^{2}-r^{2}-\bigl(x + \mfrac{d}{2}\bigr)^{2}}{(a-r)^{2}+\bigl(x + \mfrac{d}{2}\bigr)^{2}} \mathcal{E}\Bigl(k\bigl(r,x-\mfrac{d}{2}\bigr)^{2}\Bigr)\Biggr)\! \right)}
\end{aligned}\\[3ex]
\raisetag{11.5ex}
        \begin{aligned}B_{r}(r,x)= \medmath{\frac{N I \mu_{0}}{2\pi r} \! \left(\frac{x-\mfrac{d}{2}}{\sqrt{(a+r)^{2}+\bigl(x-\mfrac{d}{2}\bigr)^{2}}}\Biggl(-\mathcal{K}\Bigl(k\bigl(r,x-\mfrac{d}{2}\bigr)^{2}\Bigr) + \frac{a^{2}-r^{2} + \bigl(x-\mfrac{d}{2}\bigr)^{2}}{(a-r)^{2}+\bigl(x-\mfrac{d}{2}\bigr)^{2}} \mathcal{E}\Bigl(k\bigl(r,x-\mfrac{d}{2}\bigr)^{2}\Bigr)\Biggr) \right.}\\
\medmath{\left. -\frac{x + \mfrac{d}{2}}{\sqrt{(a+r)^{2}+\bigl(x + \mfrac{d}{2}\bigr)^{2}}}\Biggl(-\mathcal{K}\Bigl(k\bigl(r,x + \mfrac{d}{2}\bigr)^{2}\Bigr) + \frac{a^{2} + r^{2} + \bigl(x + \mfrac{d}{2}\bigr)^{2}}{(a-r)^{2}+\bigl(x + \mfrac{d}{2}\bigr)^{2}} \mathcal{E}\Bigl(k\bigl(r,x-\mfrac{d}{2}\bigr)^{2}\Bigr)\Biggr)\! \right)}
    \end{aligned}
    \end{gather}
\end{fleqn}
\end{subequations}

\end{document}