How to split long expression inside left brace?

Question

Please find below the equation I want to write on my Latex document but unfortunately i can't split the 2 expressions inside the left brace because they are too long

\begin{equation}
\begin{split}
&F\left(\lambda,\alpha,\beta,c\right) \triangleq \int_0^{\infty}\int_{-\infty}^{\frac{\lambda}{2\gamma_{rd}}} p\left(\gamma_{rd}\right)p\left(y_{rd}\mid \gamma_{rd},c\right)dy_{rd}d\gamma_{rd}= \\
&\left\lbrace
\begin{array}{ll}
1 + \displaystyle{\frac{1}{2}.\{\frac{\mu\left[c\right]\left(\exp\left(-\alpha\beta\right)-1\right) - \sqrt{B_{rd}}}{\sqrt{B_{rd}}}
+ \sqrt{\frac{\pi}{2}}.\exp\left(-\alpha\beta\right).\left(\frac{\lambda}{2\sqrt{\alpha}} - \mu\left[c\right].\sqrt{\alpha}\right).\exp\left(\iota_{\alpha,\beta}\left(\lambda\right)^{2}\right)erfc\left(\iota_{\alpha,\beta}^{+}\left(\lambda\right)\right)\}.\exp\left(\beta_{rd}^{+}\left(c\right)\lambda\right)} & \mbox{if }\lambda\geq0\\
\displaystyle{\frac{1}{2}.\{\frac{\mu\left[c\right]\left(\exp\left(-\alpha\beta\right)-1\right) + \sqrt{B_{rd}}}{\sqrt{B_{rd}}}
+ \sqrt{\frac{\pi}{2}}.\exp\left(-\alpha\beta\right).\left(\frac{\lambda}{2\sqrt{\alpha}} - \mu\left[c\right].\sqrt{\alpha}\right).\exp\left(\iota_{\alpha,\beta}\left(\lambda\right)^{2}\right)erfc\left(\iota_{\alpha,\beta}^{+}\left(\lambda\right)\right)\}.\exp\left(\beta_{rd}^{-}\left(c\right)\lambda\right)} & \mbox{if } \lambda<0
\end{array}\right.
\end{split}
\label{F function}
\end{equation}

Could you please help, I tried to put another split inside the array but without success?

Thanks in advance

Answer 1

Here's my proposal. First of all, remove almost all \left and \right, keeping only a couple of them.

Then use an inner aligned with bottom reference point for the two big functions, so they can be split in pieces.

Some notes: the low dot is never used for denoting multiplication; either nothing or \cdot (somebody uses \times when it is at the end of a line, I don't agree). Also “erfc” should be a function name, which is solved with a suitable \DeclareMathOperator.

\documentclass{article}
\usepackage{amsmath,mathtools,amssymb}

\DeclareMathOperator{\erfc}{erfc}

\begin{document}
\begin{equation}
\begin{split}
&F(\lambda,\alpha,\beta,c) \triangleq 
  \int_0^{\infty}\int_{-\infty}^{\frac{\lambda}{2\gamma_{rd}}} 
    p(\gamma_{rd})p(y_{rd}\mid \gamma_{rd},c)\,dy_{rd}\,d\gamma_{rd}= \\
&\qquad
\begin{dcases*}
\begin{aligned}[b]
1 + \frac{1}{2}
  \biggl\{
  &\frac{\mu[c](\exp(-\alpha\beta)-1) - \sqrt{B_{rd}}}{\sqrt{B_{rd}}} +{}\\
  &\sqrt{\frac{\pi}{2}}\exp(-\alpha\beta)
    \left(\frac{\lambda}{2\sqrt{\alpha}} - \mu[c]\sqrt{\alpha}\right)\cdot{}\\
  &\exp\bigl(\iota_{\alpha,\beta}(\lambda)^{2}\bigr)
    \erfc\bigl(\iota_{\alpha,\beta}^{+}(\lambda)\bigr)\biggl\}
    \exp\bigl(\beta_{rd}^{+}(c)\lambda\bigr)
\end{aligned} & if $\lambda\geq0$
\\[4ex]
\begin{aligned}[b]
\frac{1}{2}\biggl\{
  &\frac{\mu[c](\exp(-\alpha\beta)-1) + \sqrt{B_{rd}}}{\sqrt{B_{rd}}} +{}\\
  &\sqrt{\frac{\pi}{2}}\exp(-\alpha\beta)
    \left(\frac{\lambda}{2\sqrt{\alpha}} - \mu[c]\sqrt{\alpha}\right)\cdot{}\\
  &\exp\bigl(\iota_{\alpha,\beta}(\lambda)^{2}\bigr)
    \erfc\bigl(\iota_{\alpha,\beta}^{+}(\lambda)\bigr)\biggl\}
    \exp\bigl(\beta_{rd}^{-}(c)\lambda\bigr)
\end{aligned} & if $\lambda<0$
\end{dcases*}
\end{split}
\label{F function}
\end{equation}
\end{document}

Answer 2

egreg already fixed the actual problem. However, in my humble opinion, if it is going to be ugly at least you should make it worth it.

So I would propose to type it in the following format such that the reader at the very least would have the chance to realize what hit them;

\documentclass{article}
\usepackage{mathtools,amssymb,lipsum}
\DeclareMathOperator{\erfc}{erfc}

\begin{document}
\lipsum[1]
\begin{equation}
F(\lambda,\alpha,\beta,c) \triangleq 
  \int_0^{\infty}\int_{-\infty}^{\frac{\lambda}{2\gamma_{rd}}} 
    p(\gamma_{rd})p(y_{rd}\mid \gamma_{rd},c)\,dy_{rd}\,d\gamma_{rd}
\end{equation}
For nonnegative $\lambda$, this amounts to;
\begin{gather}
\frac{e^{(\beta_{rd}^{+}(c)\lambda)}}{2}\biggl\{
  -1+\frac{\mu[c]e^{-\alpha\beta-1}}{\sqrt{B_{rd}}} +
  \sqrt{\frac{\pi}{2}}e^{(\iota_{\alpha,\beta}(\lambda)^{2}-\alpha\beta)}
    \left(\frac{\lambda}{2\sqrt{\alpha}} - \mu[c]\sqrt{\alpha}\right)
    \erfc\bigl(\iota_{\alpha,\beta}^{+}(\lambda)\bigr)\biggl\}
\shortintertext{or for negative $\lambda$}
\frac{e^{(\beta_{rd}^{-}(c)\lambda)}}{2}\biggl\{
  1+\frac{\mu[c]e^{-\alpha\beta-1}}{\sqrt{B_{rd}}} +
  \sqrt{\frac{\pi}{2}}e^{(\iota_{\alpha,\beta}(\lambda)^{2}-\alpha\beta)}
    \left(\frac{\lambda}{2\sqrt{\alpha}} - \mu[c]\sqrt{\alpha}\right)
    \erfc\bigl(\iota_{\alpha,\beta}^{+}(\lambda)\bigr)\biggl\}
\end{gather}
\end{document}