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In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides an algorithm for computing the antiderivative of a rational function. In symbols, one can use ''partial fraction expansion'' to change a rational fraction in the form : where ''ƒ'' and ''g'' are polynomials, into an expression of the form : where ''g''''j'' (''x'') are polynomials that are factors of ''g''(''x''), and are in general of lower degree. Thus, the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of rational fractions, which produces a single rational fraction with a numerator and denominator usually of high degree. The ''full'' decomposition pushes the reduction as far as it can go: in other words, the factorization of ''g'' is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that fraction as a sum of a polynomial and one of several fractions, such that: * the denominator of each fraction is a power of an irreducible (not factorable) polynomial and * the numerator is a polynomial of smaller degree than this irreducible polynomial. As factorization of polynomials may be difficult, a coarser decomposition is often preferred, which consists of replacing factorization by square-free factorization. This amounts to replace "irreducible" by "square-free" in the preceding description of the outcome. == Basic principles == Assume a rational function in one indeterminate ''x'' has a denominator that factors as : over a field ''K'' (we can take this to be real numbers, or complex numbers). Assume further that ''P'' and ''Q'' have no common factor. By Bézout's identity for polynomials, there exist polynomials ''C''(''x'') and ''D''(''x'') such that : Thus and hence ''R'' may be written as : where all numerators are polynomials. Using this idea inductively we can write ''R''(''x'') as a sum with denominators powers of irreducible polynomials. To take this further, if required, write: : as a sum with denominators powers of ''F'' and numerators of degree less than ''F'', plus a possible extra polynomial. This can be done by the Euclidean algorithm, polynomial case. The result is the following theorem: . If deg ''ƒ'' < deg ''g'', then ''b'' Therefore, when the field ''K'' is the complex numbers, we can assume that each ''p''''i'' has degree 1 (by the fundamental theorem of algebra) the numerators will be constant. When ''K'' is the real numbers, some of the ''p''''i'' might be quadratic, so, in the partial fraction decomposition, a quotient of a linear polynomial by a power of a quadratic might occur. In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the ''p''''i'' may be the factors of the square-free factorization of ''g''. When ''K'' is the field of the rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor to compute the partial fraction decomposition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「partial fraction decomposition」の詳細全文を読む スポンサード リンク
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