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\section*{\LaTeX Documents on the Web}
In this document I have several math expressions and formulas
of increasing complexity.  You can see how several alternative
methods work to turn the math into usable Web materials.

\paragraph{Case 1}
In the simplest case, we may have some ''in-line'' math expressions.
For example, a sentence may have a simple equation like
\(c^2 = a^2 + b^2\) or perhaps some Greek characters like
\(2\pi r^2\).  \LaTeX is able to handle inline equations like
\(\frac{\sin\theta}{\csc\theta}+\frac{\cos\theta}{\sec\theta}=1 \).
This equation contained fractions, which forced \LaTeX to 
increase the spacing between the lines.

\paragraph{Case2}
Almost as easy, we can set the math material into a separate
''display''.
\[ \sigma^2 = \sum x^2 p(x) - \mu^2 \]

Or maybe something that takes up a little more verical space:
\[ \frac{\frac{a}{x-y} + \frac{b}{x+y}}
{1+ \frac{a-b}{a+b}} \]

Or that uses a special notation like root the operator:
\[ \sqrt[n]{\frac{x^n - y^n}{1 + u^{2n}}} \]

\paragraph{Case 3}
There are a lot more characters in the \LaTeX language than you
will find in the \emph{symbol font} set.  For example, there are
the set operators:
\[ P(A\cup B \cup C) = P(a) + P(B) + P(C) -P(A \cap B) - P(A \cap C)
  + P (B \cap C) - P(A \cap B \cap C) \]
and the vectors:
\[ \vec{\imath} + \vec{\jmath} = \vec{z} \]
Notice how the letters \emph{i} and \emph{j} are printed without
their dots when used with a vector.  Finally, here is a formula
with an overbrace and underbrace:
\[ \underbrace{a + \overbrace{b + \cdots + y}^{123} + z}_{\alpha\beta\gamma} \]

\paragraph{Case 4}
Now lets get on to the really tough stuff \ldots expressions which require
several lines of notation which lines up:
\[ y = \left\{ \begin{array} {r@{\quad:\quad}l}
        -1 & x<0 \\ 0 & x=0 \\ +1 & x>0 \end{array} \right. \]
And:
\begin{eqnarray*}
(x+y)(x-7) & = & x^2 - xy+xy-y^2 \\
 & = & x^2 - y^2 \\
(x+y)^2 & = & x^2 +2xy + y^2
\end{eqnarray*}
Finally:
\begin{eqnarray*}
 \lefteqn{1 + \frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \frac{1}{11} + \cdots}\\
 & = & \int_0^1 (1 + x^2 -x^4 -x^6 + x^8 + x^{10} - x^{12} -x^{14} +x^{16} +\cdots) dx\\
 & = & \int_0^1 \frac{1+x^2}{1+x^4}dx \\
 & = & \frac{\pi}{4}\sqrt{2}
\end{eqnarray*}
Thats all I can stand to type!
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