“partial fractions”的概念、定义、习题解题方法及翻译-数学辞典

单词

partial fractions

释义

partial fractions

Suppose that f(x)/g(x) defines a rational function, so that f(x) and g(x) are polynomials, and suppose that the degree of f(x) is less than the degree of g(x). For real polynomials, g(x) can be factorized into a product of linear factors and irreducible quadratic factors. Then the original expression f(x)/g(x) can be written as a sum of terms: corresponding to each (x−α)ⁿ in g(x), there are terms

and corresponding to each irreducible factor ax2 + bx + c in g(x), there are terms

where the scalars Ak, Bk, Ck are uniquely determined. The expression f(x)/g(x) is then said to have been written in partial fractions. If the degree of f(x) is greater than that of g(x), then long division of g(x) into f(x) can be done first to reduce the numerator to less than that of the denominator.

As examples:

The values for the numbers A, B, C,… can be found by multiplying both sides of the equation by the denominator g(x). In the last example, this gives

This has to hold for all values of x, so the coefficients of corresponding powers of x on the two sides can be equated, and this determines the unknowns. In some cases, setting x equal to particular values (in this example, x = 1) may determine some of the unknowns more quickly. The method of partial fractions is useful in the integration of rational functions or to express rational functions as power series.

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