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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 ...
Every Laurent polynomial can be written as a rational function while the converse is not necessarily true, i.e., the ring of Laurent polynomials is a subring of the rational functions. The rational function () = is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the ...
Technically, a point z 0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic (i.e. complex differentiable) in some neighbourhood of z 0. A function f is meromorphic in an open set U if for every point z of U there is a neighborhood of z in which at least one of f and 1/f is holomorphic.
Down the Rabbit Hole: ... For each s, this function gives an infinite sum, which takes some basic calculus to approach for even the simplest values of s. ... All rational numbers, and roots of ...
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 ...
Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function , whose graph is a hyperbola, and whose domain is the whole real line except for 0.
For a torus, the first Betti number is b 1 = 2 , which can be intuitively thought of as the number of circular "holes" 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 zeta function ζ(X, s) of X is by definition (,) = (=) where N m is the number of points of X defined over the degree m extension F q m of F q. The Weil conjectures state: 1. (Rationality) ζ(X, s) is a rational function of T = q −s.