Meromorphic functionIn complex analysis, a meromorphic function on an open subsetD{\displaystyle D} of the complex plane is a function that is holomorphic on all of D{\displaystyle D}except for a set of isolated points, which are poles of the function. Every meromorphic function on D{\displaystyle D} can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D{\displaystyle D}, where the poles are the zeros of the denominator.
Holomorphic functionIn mathematics, a holomorphic function is a complex-valued function of one or morecomplex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate spaceCn{\displaystyle \mathbb {C} ^{n}}. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic).
Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued functions of one or more complex variables. It is used in many branches of mathematics, including functional analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory.
Laurent seriesIn mathematics, the Laurent series of a complex functionf(z){\displaystyle f(z)} is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. Defin…
Isolated pointIn mathematics, a pointx is called an isolated point of a subset S (in a topological spaceX) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton{x} is an open set in the topological space S (considered as a subspace of X).
Complex numberNumber with a real and an imaginary partA complex number z = a + bi can be visually represented as a pair of numbers (a, b) forming a position vector (blue) or a point (red) on an Argand diagram, representing the complex plane.
General topologyIn mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
Open setIn general topology and mathematical analysis, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).
Disk (mathematics)In geometry, a disk (also spelleddisc) is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not. For a radius r{\displaystyle r}, an open disk is usually denoted as Dr{\displaystyle D_{r}}, and a closed disk is Dr¯{\displaystyle {\overline {D_{r}}}}.
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Sets cannot be mathematically defined, since they form a primitive notion. Instead, they are characterized by basic properties (axioms) that sets must have to correspond to intuition about collections.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.
Singularity (mathematics)In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the reciprocal functionf(x)=1/x{\displaystyle f(x)=1/x} has a singularity at x=0{\displaystyle x=0}, where the value of the function is not defined, as involving a division by zero.*As an Amazon Associate I earn from qualifying purchases.