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This is an equation environment in which to test fractions.  Not great translation
but at least some.   This uses \verb#\frac#
$$
\frac{1}{2} = \frac{\alpha+\beta}{\gamma+\delta}
$$
This is an equation environment in which to test fractions.  Not great translation
but at least some.  This uses \verb#\over#
$$
{1 \over 2} = {\alpha+\beta \over \gamma+\delta}
$$
Now I will test the square root.  First a simple square root
$$
z = \sqrt{x+y}
$$
now something a bit harder
$$
z = \sqrt{x^2+y^2}
$$
and what the heck, lets put a square root together with a fraction
$$
z = \frac{1}{\sqrt{x+y}}
$$
Fractions of the form
\[ \frac{\mbox{\textit{numerator}}}{\mbox{\textit{denominator}}} \]
are obtained in \LaTeX\ using the construction (This was once a bug.)

Now for some integrals.  How about a somewhat complicated one to start
$$
\int_{0}^{\infty} \frac{a}{\sqrt{x^2+c}}\,dx
$$
and
$$
\int_0^\infty \frac{a}{\sqrt{x^2+c}}\,dx
$$
and now using \verb#\limits#
$$
\int \limits_0^\infty \frac{a}{\sqrt{x^2+c}}\,dx
$$
A summation example
$$
\sum_{n=0}^\infty \frac{a_n}{\sqrt{x_n^2+c_n}}\,dx
$$
A product example
$$
\sum_{n=0}^\infty \frac{a_n}{\sqrt{x_n^2+c_n}}\,dx
$$
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