blob: b5c8b96da2888c3e23a5efef27ff7680a76fe892 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
|
%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}
}
}
|