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Given a holomorphic function f on the blue compact set and a point in each of the holes, one can approximate f as well as desired by rational functions having poles only at those three points. In complex analysis , Runge's theorem (also known as Runge's approximation theorem ) is named after the German mathematician Carl Runge who first proved ...
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 .
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:
Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions [citation needed]. The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer ...
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.
Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes:
For example, the Inverse Problem of Nevanlinna theory consists in constructing meromorphic functions with pre-assigned deficiencies at given points. This was solved by David Drasin in 1976. [ 9 ] Another direction was concentrated on the study of various subclasses of the class of all meromorphic functions in the plane.
Rational points can be directly characterized by height functions which measure their arithmetic complexity. [ 5 ] The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry.
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