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In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
When a partial fraction term has a single (i.e. unrepeated) binomial in the denominator, the numerator is a residue of the function defined by the input fraction. We calculate each respective numerator by (1) taking the root of the denominator (i.e. the value of x that makes the denominator zero) and (2) then substituting this root into the ...
In complex analysis, a partial fraction expansion is a way of writing a meromorphic function as an infinite sum of rational functions and polynomials. When f ( z ) {\displaystyle f(z)} is a rational function, this reduces to the usual method of partial fractions .
A continued fraction is an expression of the form = + + + + + where the a n (n > 0) are the partial numerators, the b n are the partial denominators, and the leading term b 0 is called the integer part of the continued fraction.
For example, consider the following rational function, along with its partial fraction expansion: = () = + (). This function has singularities at z = 1 {\displaystyle z=1} and z = 2 i {\displaystyle z=2i} , where the denominator is zero and the expression is therefore undefined.
One possible proof outline is as follows. If is finite, it suffices to take () = ().If is not finite, consider the finite sum () = where is a finite subset of .While the () may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of (provided by Runge's theorem) without changing the principal parts of the () and in such a way that convergence is ...