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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. [1]
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 ...
Algebraic factoring of expressions, including partial fraction decomposition. Algebraic simplification; for example, the CAS can combine multiple terms into one fraction by finding a common denominator. Evaluation of trigonometric expressions to exact values. For example, sin(60°) returns instead of 0.86603.
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 .
The partial fraction decomposition is given by ... A Lucas Number Calculator can be found here. OEIS sequence A000032 (Lucas numbers beginning at 2) This page was ...
This is useful in solving such recurrences, since by using partial fraction decomposition we can write any proper rational function as a sum of factors of the form 1 / (ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.
A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals + to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that
Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. The theorem is named after the Swedish mathematician Gösta Mittag-Leffler who published versions of the theorem in 1876 and ...