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Mathematics-Online course: Prepcourse Mathematics - Analysis - Functions

Rational Function

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A rational function $ r$ with the degree of the nominator $ m$ and the degree of the denominator $ n$ is the quotient of two polynomials:

$\displaystyle r(x) = \frac{p(x)}{q(x)} = \frac{a_0+a_1x+\cdots+a_mx^m}{b_0+b_1x+\cdots+b_nx^n} \,. $

This representation is irreducible if $ p$ and $ q$ have no common linear factor.

Then the zeros of the denominator are called poles. The order of the pole corresponds to the multiplicity of the zero.

The function

$\displaystyle f(x)=\frac{x^4}{(x+1)(x-1)^2}$

has a single pole at $ x=-1$ as well as a double pole at $ x=1$.

\includegraphics[width=7.4cm]{xxxx_1+x_1-x_1-x.eps}

The illustration shows that $ f$ changes its algebraic sign at a single pole whereas a double pole retains the algebraic sign.

(Authors: App/Höllig/Abele)
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  automatically generated 9/18/2007