diff options
Diffstat (limited to 'info/mathtrip/src/partial.tex')
-rw-r--r-- | info/mathtrip/src/partial.tex | 44 |
1 files changed, 44 insertions, 0 deletions
diff --git a/info/mathtrip/src/partial.tex b/info/mathtrip/src/partial.tex new file mode 100644 index 0000000000..b5c8b96da2 --- /dev/null +++ b/info/mathtrip/src/partial.tex @@ -0,0 +1,44 @@ +%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} + } +} |