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The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers ; they may be taken in any field K .
A generalization to the matrix case (matrices with polynomial function entries that are always positive semidefinite can be expressed as sum of squares of symmetric matrices with rational function entries) was given by Gondard, Ribenboim [13] and Procesi, Schacher, [14] with an elementary proof given by Hillar and Nie.
Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite x values. Thus, rational functions can easily be incorporated into a rational function model. Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator.
Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
The sheaf of rational functions K X of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties , such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, K X ( U ) is the ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
In complex analysis, a partial fraction expansion is a way of writing a meromorphic function as an infinite sum of rational functions and polynomials. When f ( z ) {\displaystyle f(z)} is a rational function, this reduces to the usual method of partial fractions .