“partial fractions”的概念、定义、翻译、参考文献-科学参考

partial fractions

Mathematics

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

$\frac{A_{1}}{x - α} + \frac{A_{2}}{{(x - α)}^{2}} + \dots + \frac{A_{n}}{{(x - α)}^{n}},$

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

$\frac{B_{1} x + C_{1}}{a x^{2} + b x + c} + \frac{B_{2} x + C_{2}}{{(a x^{2} + b x + c)}^{2}} + \dots + \frac{B_{n} x + C_{n}}{{(a x^{2} + b x + c)}^{n}},$

where the scalars A_k, B_k, C_k 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:

$\begin{matrix} \frac{3}{(x - 1) (x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}, \\ \frac{3 x^{2} + 2 x + 1}{{(x - 1)}^{3}} = \frac{A}{x - 1} + \frac{B}{{(x - 1)}^{2}} + \frac{C}{{(x - 1)}^{3}}, \\ \frac{3 x + 2}{(x - 1) {(x^{2} + x + 1)}^{2}} \\ = \frac{A}{x - 1} + \frac{B x + C}{x^{2} + x + 1} + \frac{D x + E}{{(x^{2} + x + 1)}^{2}}, \\ \frac{3 x + 2}{{(x - 1)}^{2} (x^{2} + x + 1)} = \frac{A}{x - 1} + \frac{B}{{(x - 1)}^{2}} + \frac{C x + D}{x^{2} + x + 1} . \end{matrix}$

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

$3 x + 2 = A (x - 1) (x^{2} + x + 1) + B (x^{2} + x + 1) + (C x + D) {(x - 1)}^{2} .$

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.

Chemical Engineering

A set of simple fractions from which a more complicated fraction can be resolved.

单词	partial fractions
释义	partial fractions Mathematics 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 $\frac{A_{1}}{x - α} + \frac{A_{2}}{{(x - α)}^{2}} + \dots + \frac{A_{n}}{{(x - α)}^{n}},$ and corresponding to each irreducible factor ax² + bx + c in g(x), there are terms $\frac{B_{1} x + C_{1}}{a x^{2} + b x + c} + \frac{B_{2} x + C_{2}}{{(a x^{2} + b x + c)}^{2}} + \dots + \frac{B_{n} x + C_{n}}{{(a x^{2} + b x + c)}^{n}},$ where the scalars A_k, B_k, C_k 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: $\begin{matrix} \frac{3}{(x - 1) (x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2}, \\ \frac{3 x^{2} + 2 x + 1}{{(x - 1)}^{3}} = \frac{A}{x - 1} + \frac{B}{{(x - 1)}^{2}} + \frac{C}{{(x - 1)}^{3}}, \\ \frac{3 x + 2}{(x - 1) {(x^{2} + x + 1)}^{2}} \\ = \frac{A}{x - 1} + \frac{B x + C}{x^{2} + x + 1} + \frac{D x + E}{{(x^{2} + x + 1)}^{2}}, \\ \frac{3 x + 2}{{(x - 1)}^{2} (x^{2} + x + 1)} = \frac{A}{x - 1} + \frac{B}{{(x - 1)}^{2}} + \frac{C x + D}{x^{2} + x + 1} . \end{matrix}$ 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 $3 x + 2 = A (x - 1) (x^{2} + x + 1) + B (x^{2} + x + 1) + (C x + D) {(x - 1)}^{2} .$ 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. Chemical Engineering A set of simple fractions from which a more complicated fraction can be resolved.
随便看	Audubon, John James Auer Aufbau principle aufheben Aufklärung augen-gneiss auger Auger effect Auger electron Auger electron spectroscopy Auger, Pierre Victor Auger process Auger shower Auger spectroscopy augite augite-minette augmentation augmented addressing augmented Dickey-Fuller augmented Dickey–Fuller test augmented matrix augmented Phillips curve augmented reality augmented transition network Augsburg Confession