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Diffstat (limited to 'source/know/concept/partial-fraction-decomposition')
| -rw-r--r-- | source/know/concept/partial-fraction-decomposition/index.md | 20 |
1 files changed, 10 insertions, 10 deletions
diff --git a/source/know/concept/partial-fraction-decomposition/index.md b/source/know/concept/partial-fraction-decomposition/index.md index 03c1c76..bb7faa2 100644 --- a/source/know/concept/partial-fraction-decomposition/index.md +++ b/source/know/concept/partial-fraction-decomposition/index.md @@ -8,15 +8,15 @@ layout: "concept" --- **Partial fraction decomposition** or **partial fraction expansion** -is a method to rewrite quotients of two polynomials $g(x)$ and $h(x)$, -where the numerator $g(x)$ is of lower order than $h(x)$, -as sums of fractions with $x$ in the denominator: +is a method to rewrite quotients of two polynomials $$g(x)$$ and $$h(x)$$, +where the numerator $$g(x)$$ is of lower order than $$h(x)$$, +as sums of fractions with $$x$$ in the denominator: $$\begin{aligned} f(x) = \frac{g(x)}{h(x)} = \frac{c_1}{x - h_1} + \frac{c_2}{x - h_2} + ... \end{aligned}$$ -Where $h_n$ etc. are the roots of the denominator $h(x)$. If all $N$ of +Where $$h_n$$ etc. are the roots of the denominator $$h(x)$$. If all $$N$$ of these roots are distinct, then it is sufficient to simply posit: $$\begin{aligned} @@ -25,8 +25,8 @@ $$\begin{aligned} } \end{aligned}$$ -The constants $c_n$ can either be found the hard way, -by multiplying the denominators around and solving a system of $N$ +The constants $$c_n$$ can either be found the hard way, +by multiplying the denominators around and solving a system of $$N$$ equations, or the easy way by using this trick: $$\begin{aligned} @@ -35,7 +35,7 @@ $$\begin{aligned} } \end{aligned}$$ -If $h_1$ is a root with multiplicity $m > 1$, then the sum takes the form of: +If $$h_1$$ is a root with multiplicity $$m > 1$$, then the sum takes the form of: $$\begin{aligned} \boxed{ @@ -44,15 +44,15 @@ $$\begin{aligned} } \end{aligned}$$ -Where $c_{1,j}$ are found by putting the terms on a common denominator, e.g. +Where $$c_{1,j}$$ are found by putting the terms on a common denominator, e.g. $$\begin{aligned} \frac{c_{1,1}}{x - h_1} + \frac{c_{1,2}}{(x - h_1)^2} = \frac{c_{1,1} (x - h_1) + c_{1,2}}{(x - h_1)^2} \end{aligned}$$ -And then, using the linear independence of $x^0, x^1, x^2, ...$, solving -a system of $m$ equations to find all $c_{1,1}, ..., c_{1,m}$. +And then, using the linear independence of $$x^0, x^1, x^2, ...$$, solving +a system of $$m$$ equations to find all $$c_{1,1}, ..., c_{1,m}$$. |