Split environment wrong braces dimensions

Question

I have to split a quite long formula. I used a nested split environment but I have to break curly braces from one line to another using \right. and \left. The curly braces I get have different dimensions. I tried using \Biggl and \Biggr but they are too small. Here is my source:

\begin{equation}
  \label{eq:splitted}
\begin{split}
  c(x, y, t) &= \frac{M/Y}{b\sqrt{4\pi\left(\D_x^T+\K_x\right)t}}
  \exp\left[-\frac{\left(x-Ut-x_0\right)^2}{4\left(\D_x^T+\K_x\right)t}\right]\\
  &\begin{split}\frac{b}{\sqrt{4\pi\left(\D_y^T+\K_y\right)t}}
  \left\{
\sum_{j=-\infty}^{\infty}%
\exp\left[-\frac{\left(y-y_0+2jb\right)^2}{4\left(\D_y^T+\K_y\right)}\right]\right.&%
  \\ + \left.%
    \exp\left[-\frac{\left(y+y_0+2jb\right)^2}{4\left(\D_y^T+\K_y\right)}\right]\right\}&
  \end{split}
  \end{split}
\end{equation}

Answer 1

No need of \phantom and split; the best approach in this case is with the multlined environment provided by mathtools.

\documentclass{article}
\usepackage{mathtools}
\newcommand{\D}{\mathbf{D}}
\newcommand{\K}{\mathbf{K}}

\begin{document}
\begin{equation}\label{eq:split}
\begin{gathered}
  c(x, y, t) =
\begin{multlined}[t]
  \frac{M/Y}{b\sqrt{4\pi(\D_x^T+\K_x)t}}
  \exp\biggl[-\frac{(x-Ut-x_0)^2}{4(\D_x^T+\K_x)t}\biggr]\\
  \frac{b}{\sqrt{4\pi(\D_y^T+\K_y)t}}
  \biggl\{
    \sum_{j=-\infty}^{\infty}\exp\biggl[-\frac{(y-y_0+2jb)^2}{4(\D_y^T+\K_y}\biggr]\\
    {}+\exp\biggl[-\frac{(y+y_0+2jb)^2}{4(\D_y^T+\K_y)}\biggr]
  \biggr\}
\end{multlined}
\end{gathered}
\end{equation}
\end{document}

Notice that I've removed all \left and \right before the inner parentheses: they aren't needed and add spacing.

By choosing \biggl[ and \biggr] instead of \left[ and \right] we get uniform height of the brackets and we can use also the same size for the braces. In general, when a summation has limits above and below, \biggl and \biggr are to be preferred to \left and \right that produce too large symbols.

The wrapping gathered environment is used for getting the equation number centered with respect to the whole formula.

---

A refinement can be obtained by seeing that the two square root symbols are different though they contain similar formulas. If the second one is typed as

\sqrt{\smash[b]{4\pi(\D_y^T+\K_y)t}\vphantom{_x}}

the result will be

Answer 2

First of all, it is not a good idea to nest the split environment, and it does not seem necessary in your code. To fix the brace size you can insert an invisible sum in the last line via \vphantom that forces the closing brace to be higher:

\begin{equation}
  \label{eq:splitted}
\begin{split}
  c(x, y, t) &= \frac{M/Y}{b\sqrt{4\pi\left(\D_x^T+\K_x\right)t}}
  \exp\left[-\frac{\left(x-Ut-x_0\right)^2}{4\left(\D_x^T+\K_x\right)t}\right] \\
  & \phantom{{}={}} \frac{b}{\sqrt{4\pi\left(\D_y^T+\K_y\right)t}}
  \left\{ \sum_{j=-\infty}^{\infty}  \exp\left[-\frac{\left(y-y_0+2jb\right)^2}{4\left(\D_y^T+\K_y\right)}\right]\right.  \\ 
  & \phantom{{}={}} + \left. \vphantom{\sum_{j=-\infty}^{\infty}} \exp\left[-\frac{\left(y+y_0+2jb\right)^2}{4\left(\D_y^T+\K_y\right)}\right]\right\}
  \end{split}
\end{equation}

I also added invisible = after the alignment characters in the second and third line, so everything is right to the =.

To get the last line flushed right, you can nest a multlined from package mathtools in it:

\usepackage{mathtools}
...
\begin{equation}
  \label{eq:splitted2}
\begin{split}
  c(x, y, t) &= \frac{M/Y}{b\sqrt{4\pi\left(\D_x^T+\K_x\right)t}}
  \exp\left[-\frac{\left(x-Ut-x_0\right)^2}{4\left(\D_x^T+\K_x\right)t}\right] \\
  & \phantom{{}={}}
  \begin{multlined}
  \frac{b}{\sqrt{4\pi\left(\D_y^T+\K_y\right)t}}
  \left\{ \sum_{j=-\infty}^{\infty}  \exp\left[-\frac{\left(y-y_0+2jb\right)^2}{4\left(\D_y^T+\K_y\right)}\right]\right.  \\ 
   + \left. \vphantom{\sum_{j=-\infty}^{\infty}} \exp\left[-\frac{\left(y+y_0+2jb\right)^2}{4\left(\D_y^T+\K_y\right)}\right]\right\}
  \end{multlined}
  \end{split}
\end{equation}