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+%This command provides the math for the second horizontal area
+%of the first column of the page 6
+%
+%The command has one parameter
+% 1) The width of the math text
+\newcommand\TSixPartialFrac[1]{%
+ \parbox[t]{#1}{%
+ \TSixPartialFontSize
+ \DisplaySpace{\TSixDisplaySpace}{\TSixDisplayShortSpace}
+ %Since the columns is narrow, ragged right looks better
+ \raggedright
+
+ Let $N(x)$ and $D(x)$ be polynomial functions of $x$.
+
+ We can break down $N(x)/D(x)$ using partial fraction expansion.
+
+ First,
+ if the degree of $N$ is greater than or equal to the degree of $D$,
+ divide $N$ by $D$,
+ obtaining
+ \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
+ \Fm{\frac{N(x)}{D(x)} = Q(x) + \frac{N'(x)}{D(x)}}
+ \end{DisplayFormulae}
+ where the degree of $N'$ is less than that of $D$.
+
+ \TSixTitle{Second, factor $D(x)$}
+ Use the following rules:
+
+ \mbox{For a non-repeated factor:}
+ \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
+ \Fm{\frac{N(x)}{(x-a) D(x)} = \frac{A}{x-a} + \frac{N'(x)}{D(x)}}
+ where
+ \Fm{A = \left[\frac{N(x)}{D(x)}\right]_{x=a}}
+ \end{DisplayFormulae}
+
+ \mbox{For a repeated factor:}
+ \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
+ \Fm{\frac{N(x)}{(x-a)^m D(x)} =\sum_{k=0}^{m-1}\frac{A_k}{(x-a)^{m-k}} + \frac{N'(x)}{D(x)}}
+ where
+ \Fm{A_k = \frac{1}{k!}\left[\frac{d^k}{dx^k}
+ \left(\frac{N(x)}{D(x)}\right)\right]_{x=a}}
+ \end{DisplayFormulae}
+ }
+}