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:

    “Preaching is the expression of the moral sentiment in application to the duties of life.”
    —Ralph Waldo Emerson (1803–1882)

    “Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.”
    —Talcott Parsons (1902–1979)

    “The act of bellringing is symbolic of all proselytizing religions. It implies the pointless interference with the quiet of other people.”
    —Ezra Pound (1885–1972)

    “The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
    —Claude Lévi-Strauss (b. 1908)