Partial Fraction Partial Fraction The expansion by the Partial Fractions can be used to convert any algebraic rational function of the form p (x) / q (x) where p and q are polynomials into a function which will be of the form of summation of pj (x) / qj (x) with respect to j and where qj (x) are the polynomials which are the factors of q (x) and generally they have a lower degree. Therefore the decomposition by the partial fractions can be understood as the reverse method of the elementary addition operation of the algebraic rational fractions which gives just one rational function generally having a numerator and the denominator which have high degree. The complete decomposition forces the reduction to go as far as possible or we can say that in a different way that the factorization of q is utilized as more as it can be done. Hence, the result of the complete partial fractions shows that function as an addition of fractions in which the denominator of every single term is the power of such a polynomial which cannot be factorized and the numerator will be a polynomial of a degree which is smaller than that of the polynomial which is not reducible. The algorithm of Euclidean can be utilized to directly reduce the degree of the numerator but this algorithm is not useful if the numerator p is already having a lower degree than the denominator q.
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The main motive of decomposing an algebraic rational function into an addition of the simpler functions with the help of the partial fractions is that it makes it easier to do the linear operations on it. Hence the problems which are faced in the computation of the integrals, derivatives and antiderivatives, in the expansions of the Power series and the Fourier series and in the linear and the functional conversion of the rational functions can be decreased to a very great level with the help of the decomposition by the partial fractions. The computation can be made on every single element which is used in the process of the decomposition by the partial fractions. We can take an example of the extreme use of the partial fractions in the problems of the integration. The partial fractions can be used in the problems of the integration for calculating the antiderivatives. The field of the scalars we adopt tells us that which polynomials are not reducible. Therefore we can conclude that the degree of the polynomials which are not reducible will either be 1 or 2 if we take only the real numbers. However, if we allow the complex numbers, then only the polynomials which are of degree equal to one can be irreducible. But, some of the polynomials of the higher degree are also irreducible if we allow only the rational numbers or a field which is finite. partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function (also known as a rational algebraic fraction). In symbols, one can use partial fraction expansion to change a rational function in the form where Ć’ and g are polynomials, into a function of the form
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where gj (x) are polynomials that are factors of g(x), and are in general of lower degree. Thus the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of algebraic fractions, that produces a single rational function with a numerator and denominator usually of high degree. The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that function as a sum of fractions, where: the denominator of each term is a power of an irreducible (not factorable) polynomial and the numerator is a polynomial of smaller degree than that irreducible polynomial. To decrease the degree of the numerator directly, the Euclidean algorithm can be used, but in fact if Ć’ already has lower degree than g this isn't helpful. The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it. Therefore the problem of computing derivatives, antiderivatives, integrals, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition. Partial fractions in integration for an account of the use of the partial fractions in finding antiderivatives. Just which polynomials are irreducible depends on which field of scalars one adopts. Thus if one allows only real numbers, then irreducible polynomials are of degree either 1 or 2. If complex numbers are allowed, only 1st-degree polynomials can be irreducible. If one allows only rational numbers, or a finite field, then some higher-degree polynomials are irreducible.
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Published on Sep 1, 2012