\ProcInst{xml version="1.0" encoding="us-ascii"}
\ProcInst{xml-stylesheet type="text/xsl" href="mathml.xsl"}
\begin{html}[xmlns="http://www.w3.org/1999/xhtml"]
% Locate gellmu.el as "http://www.albany.edu/~hammond/gellmu/gellmu.el"
% Use: "emacs -batch -l gellmu.el -f gellmu-xml whatever.glm" to make
% "whatever.xml".  Then view "whatever.xml" with MathML-enabled Mozilla.
% The "mathml-altheim" \documenttype option sets it up for Murray Altheim's
%   XHTML 1.1 plus MathML 2.0 unified document type definition.
%
% The following type of "package building" is run-time inefficient,
% but it is good for test work.
%
% DTD Wars: somebody is blocking the use of DTD's so we need entity codes
%
\macro{\ }{ }
\newcommand{\copy}{©}
\newcommand{\delta}{\mi{δ}}
\newcommand{\pi}{\mi{π}}
\newcommand{\infty}{\mi{∞}}
\newcommand{\pm}{\mo{±}}
\newcommand{\times}{×}
\newcommand{\Integral}{\mo{∫}}
\newcommand{\Product}{\mo{∏}}
\newcommand{\Sum}{\mo{∑}}
%
\newcommand{\glmu}{http://www.albany.edu/~hammond/gellmu}
\newcommand{\glmv}{http://math.albany.edu:8000/~hammond/gellmu}
\newcommand{\glma}[2][]{\a[href="\glmu/#1"]{#2}}
\newcommand{\glmb}[2][]{\a[href="\glmv/#1"]{#2}}
\newcommand{\href}[2][http://www.w3.org/]{\a[href="#1"]{#2}}
\newcommand{\w3u}{http://www.w3.org}
\newcommand{\w3a}[2][]{\a[href="\w3u/#1"]{#2}}
\newcommand{\mns}{xmlns="http://www.w3.org/1998/Math/MathML"}
\newcommand{\irmath}[1]{\math[\mns mode="inline"]{\mrow{#1}}}
\newcommand{\drmath}[1]{\math[\mns class="display" mode="display"]{\mrow{#1}}}
\newcommand{\plus}{\mo{+}}
\newcommand{\minus}{\mo{-}}
\newcommand{\neg}[1]{\mrow{\minus#1}}
\newcommand{\mul}{\mo{ }}
\newcommand{\eq}{\mo{=}}
\newcommand{\eqn}[2]{\mrow{#1}\mo{=}\mrow{#2}}
\newcommand{\bal}[1]{\mfenced{\mrow{#1}}}
\newcommand{\balsb}[1]{\mfenced[open="[" close="]"%
            ]{\mrow{#1}}}
\newcommand{\balbr}[1]{\mfenced[open="\{" close="\}"%
            ]{\mrow{#1}}}
\newcommand{\balinv}[1]{\mfenced[open=" " close=" "%
            ]{\mrow{#1}}}
\newcommand{\lbalbr}[1]{\mfenced[open="\{" close=""%
            ]{\mrow{#1}}}
\newcommand{\norm}[1]{\mfenced[open="||" close="||"%
            ]{\mrow{#1}}}
\newcommand{\gexp}[2][\mi{e}]{\msup{\mrow{#1}\mrow{#2}}}
\newcommand{\sgexp}[2][\mi{e}]{\msup{#1#2}}
\newcommand{\dlog}[1]{\sfrac{\mr{\mi{d#1}}}{\mi{#1}}}
\newcommand{\ifrac}[2]{\mr{#1\mo{/}#2}}
\newcommand{\frac}[2]{\mfrac{\mrow{#1}\mrow{#2}}}
\newcommand{\sfrac}[2]{\mfrac{#1#2}}
\newcommand{\sqrt}[1]{\msqrt{\mrow{#1}}}
\newcommand{\int}[3]{\msubsup{\Integral;\mrow{#1}#2}\mrow{#3}}
\newcommand{\sum}[3]{\msubsup{\Sum;\mrow{#1}#2}\mrow{\mspace;#3}}
\newcommand{\prod}[3]{\msubsup{\Product;\mrow{#1}#2}\mrow{#3}}
\newcommand{\mxtwo}[4]{%
\mtable{\mtr{\mtd{#1}\mtd{#2}}\mtr{\mtd{#3}\mtd{#4}}}%
}
% This defn needs to appear after any use in others.
\newcommand{\mr}[1]{\mrow{#1}}
\begin{head}
\style[type="text/css"]{
body {
  html,background: #fff;
  color: black;
  background-color: white;
}

h1 {
  display: block;
  margin-left: auto;
  margin-right: auto;  
  text-align: center;
}


.display {
  display: block;
  margin-left: auto;
  margin-right: auto;
  text-align: center;
}
}
\title{MathML Examples}
\end{head}
\begin{body}
\h1{MathML Examples}
\h2[class="display"]{William F. Hammond}
\p[class="display"]{Copyright \copy; 2001 William F. Hammond}

\p{
This is an XHTML document with MathML markup prepared using the basic
layer of GELLMU and the XML namespaces regime for extending the basic
tagset of XHTML.
}
\p{
The following relation is sometimes called the \em{parallelogram law}.
}
\drmath{
 \eqn{
  \gexp{\norm{\mi{a}}}{\mn{2}}
  \plus
  \gexp{\norm{\mi{b}}}{\mn{2}}
 }{
  \frac{\mn{1}}{\mn{2}}
  \mul
  \balbr{
   \gexp{\norm{\mi{a}\plus \mi{b}}}{\mn{2}}
   \plus
   \gexp{\norm{\mi{a}\minus\mi{b}}}{\mn{2}}
  }
 }
}

\p{Balancers should be stretched when appropriate.  Here the simple
fraction
\irmath{\ifrac{\mn{1}}{\mn{2}}}
is multiplied with a complex fraction.
}
\drmath{
 \frac{\mn{1}}{\mn{2}}
 \mul
 \balbr{
  \frac{
   \frac{\mi{a}}{\mi{b}}
  }{
   \frac{\mi{c}}{\mi{d}}
  }
 } 
}

\p{This is MathML markup of the formula for the roots of the quadratic
polynomial
\irmath{\mtext{\ }
\mi{a}\mul\gexp{\mi{x}}{\mn{2}}\plus\mi{b}\mul\mi{x}\plus\mi{c}} .
}
\drmath{
 \mi{x}
 \eq
 \frac{
  \neg{\mi{b}}
  \pm
  \gexp{
   \bal{\gexp{\mi{b}}{\mn{2}}\minus\mn{4}\mul\mi{a}\mul\mi{c}}
  }{
   \mn{1}\mo{/}\mn{2}
  }
 }{
  \mn{2}\mul\mi{a}
 }
}

\p{This is MathML markup for a 2 \times; 2 matrix.}
\drmath{
\mi{A}\eq\balsb{
\mxtwo{\mi{a}}{\mi{b}}{\mi{c}}{\mi{d}}
}}

\p{Taylor's Theorem:}
\drmath{
 \eqn{
  \mo{f}\bal{\mi{x}}
 }{
 \sum{
   \eqn{\mi{j}}{\mn{0}}
  }{
   \infty
  }{
   \balinv{
    \frac{\gexp{\mo{f}}{\bal{\mi{j}}}\bal{\mi{0}}}{\mi{j}\mo{!}}
    \mul
    \gexp{\mi{x}}{\mi{j}}
   }
  }
 }
}

\p{This is a form of the Weierstrass infinite product expansion of the
gamma function.}
\drmath{
 \mrow{
  \int{\mn{0}}{\infty}{
   \gexp{\mi{t}}{\mi{x}} \mul \gexp{\mi{-t}} \mul
   \dlog{t}
  }
 }
 \eq
 \balinv{
  \frac{\mn{1}}{\mi{x}} \mul
  \prod{\mrow{\mi{k}\mo{=}\mn{1}}}{\infty}{
   \mrow{
    \frac{
     \gexp{\bal{\mn{1}\plus\frac{\mn{1}}{\mi{k}}}}{\mi{x}}
    }{
     \bal{\mn{1}\plus\frac{\mi{x}}{\mi{k}}}
    }
   }
  }
 }
}

\p{
Dirac's \irmath{\delta;}-function, which is actually a distribution in the
sense of L. Schwartz rather than a function, is characterized by the
property that for every \irmath{\msup{\mi{C}\infty}}
function \irmath{\mi{f}} with compact support one has:
}
\drmath{
 \eqn{
  \int{\neg{\infty}}{\infty}{\mi{f}\delta;}
 }{
  \mo{f}\bal{\mn{0}}
 }
 \mtext{\ .}
}
\p{In particular, when \irmath{\mi{f}} is the characteristic function
\irmath{\msub{\mi{I}\mi{S}}} of a set \irmath{\mi{S}}:
}
\drmath{
 \eqn{
  \int{\mi{S}}{\mi{}}{\delta;}
 }{
  \lbalbr{
   \mtable{
    \mtr{
     \mtd{\mn{1}}
     \mtd{\mtext{\ if\ }\mi{S}\mtext{\ contains\ }\mn{0}\mtext{.}}
    }
    \mtr{
     \mtd{\mn{0}}
     \mtd{\mtext{\ otherwise.}}
    }
   }
  }
 }
}
\end{body}
\end{html}