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:

    “The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
    —Jorge Luis Borges (1899–1986)

    “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)

    “The more specific idea of evolution now reached is—a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter.”
    —Herbert Spencer (1820–1903)