Partial Fraction - Application To Symbolic Integration

Application To Symbolic Integration

For the purpose of symbolic integration, the preceding result may be refined into

Let ƒ and g be nonzero polynomials over a field K. Write g as a product of powers of pairwise coprime polynomials which have no multiple root in an algebraically closed field:

There are (unique) polynomials b and c ij with deg c ij < deg p i such that
\frac{f}{g}=b+\sum_{i=1}^k\sum_{j=2}^{n_i}\left(\frac{c_{ij}}{p_i^{j-1}}\right)' + \sum_{i=1}^k \frac{c_{i1}}{p_i}.
where denotes the derivative of

This reduces the computation of the antiderivative of a rational function to the integration of the last sum, with is called the logarithmic part, because its antiderivative is a linear combination of logarithms.

Read more about this topic:  Partial Fraction

Famous quotes containing the words application to, application, symbolic and/or integration:

    ““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “The main object of a revolution is the liberation of man ... not the interpretation and application of some transcendental ideology.”
    —Jean Genet (1910–1986)

    “I find it profoundly symbolic that I am appearing before a committee of fifteen men who will report to a legislative body of one hundred men because of a decision handed down by a court comprised of nine men—on an issue that affects millions of women.... I have the feeling that if men could get pregnant, we wouldn’t be struggling for this legislation. If men could get pregnant, maternity benefits would be as sacrosanct as the G.I. Bill.”
    —Letty Cottin Pogrebin (20th century)

    “Look back, to slavery, to suffrage, to integration and one thing is clear. Fashions in bigotry come and go. The right thing lasts.”
    —Anna Quindlen (b. 1952)