What is the purpose of "\left.", i.e., "\left" followed by a period?

Question

What is the purpose of \left., i.e., \left followed by a period? Similarly for \right.

E.g. for \right. and \left (no period)

begin{eqnarray}\label{Dsys}
  \textcolor[rgb]{.25,.0,.8}{%
    \left\{\begin{array}{l}
              u_t=F(u) \\ \\ 
              u(0)=u_0\in H,
           \end{array}
     \right. 
  }
\end{eqnarray}

Answer 1

I assume you're familiar with the most common use case for \left and \right, which is to provide automatic sizing of visual delimiters or "fences" -- round parentheses, square brackets, curly braces, angle-brackets, vertical bars, etc -- so that the "fences" may visually enclose the material they surround. Example:

\[ \left( \frac{a}{b} \right) \]

Well, sometimes a "one-sided fence" is called for. The example code you provided is such a case: we need a large curly brace to the left of the array, but nothing to the right of the array. To cater to such needs, TeX lets you "pair" the \left\{ statement with a \right. statement. If you will, the "." symbol after \right denotes "no fence on this side". Because \left and \right must occur in pairs and need to operate on something, it wouldn't work to write \right without a suitable argument. (As explained below, LaTeX issues a warning message if no suitable argument is encountered.) Similarly, there may be cases for which one needs to "pair" a \left. directive with a \right| directive, as in

\[ \left. \frac{\partial f}{\partial x} \right|_{x=x_0} \]

Aside: In case you're wondering how TeX goes about processing the material that immediately follows \left and \right, here's a TeXnical explanation. As @egreg has pointed out in a comment, TeX assigns a "delimiter code" (\delcode for short) to all (math-mode) tokens: "fence symbols" -- such as round parentheses, square brackets, curly braces, and vertical bars -- have a positive \delcode, the . ("dot", "period", "full stop") symbol has a \delcode of zero, and all other tokens have -1 as their \delcode. If the token that follows \left or \right has a positive \delcode, i.e., if a "real" fence symbol is to be processed, the fence is sized according to TeX's algorithms; if the \delcode is zero, nothing is done (apart from the insertion of a bit of horizontal whitespace); and if the \delcode is negative, TeX issues a

Missing delimiter (. inserted)

warning message. This message should be taken seriously, i.e., one should examine the code and fix it appropriately. For more information on \delcodes, do peruse Chapter 17 of The TeXbook, entitled "More about Math", and especially the second double-dangerous-bend material on p. 156. For still more information, please check out pages 290 and 345 of The TeXbook.

Incidentally, the amsmath package provides an environment called cases which is designed to typeset your example code in a way that focuses on the meaning of what you're writing, without distractions about how one should go about implementing the typesetting chore.

\documentclass{article}
\usepackage{amsmath,xcolor}
\begin{document}
\begin{equation}\label{Dsys}
\textcolor[rgb]{.25,.0,.8}{%
   \begin{cases}
       u_t=F(u)    \\ 
       u(0)=u_0\in H
   \end{cases} 
}
\end{equation}
\end{document}