Domain of holomorphy. The sets in the definition. In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a set which is maximal in the sense that there exists a holomorphic function on this set which cannot be extended to a bigger set. Formally, an open set in the n-dimensional complex space is called a domain of holomorphy if there do not exist non-empty open sets and where is connected, such that for every holomorphic function on there exists a holomorphic function with In the this is no longer true, as it follows from Hartogs' lemma.
Equivalent conditions[edit] For a domain the following conditions are equivalent: Implications are standard results (for , see Oka's lemma) . , i.e. constructing a global holomoprhic function which admits no extension from non-extendable functions defined only locally. -problem). Properties[edit] References[edit] Steven G. See also[edit]
Complex number. A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i2 = −1. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation x2 = −1, that is, i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point (a, b) in the complex plane. Overview[edit] Complex numbers allow for solutions to certain equations that have no solutions in real numbers. Definition[edit] . Or or z*. and . Linear complex structure. In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space. Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds.
The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a "linear complex structure". Definition and properties[edit] A complex structure on a real vector space V is a real linear transformation such that J2 = −idV. Here J2 means J composed with itself and idV is the identity map on V. ). Cn[edit] Complexification. Formal definition[edit] Let V be a real vector space. The complexification of V is defined by taking the tensor product of V with the complex numbers (thought of as a two-dimensional vector space over the reals): The subscript R on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted).
As it stands, VC is only a real vector space. More generally, complexification is an example of extension of scalars – here extending scalars from the real numbers to the complex numbers – which can be done for any field extension, or indeed for any morphism of rings. Formally, complexification is a functor VectR → VectC, from the category of real vector spaces to the category of complex vector spaces. Basic properties[edit] By the nature of the tensor product, every vector v in VC can be written uniquely in the form where v1 and v2 are vectors in V.
Where or defined by. Complex geometry. In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis. Throughout this article, "analytic" is often dropped for simplicity; for instance, subvarieties or hypersurfaces refer to analytic ones.
Following the convention in Wikipedia, varieties are assumed to be irreducible. Definitions[edit] An analytic subset of a complex-analytic manifold M is locally the zero-locus of some family of holomorphic functions on M. Line bundles and divisors[edit] Throughout this section, X denotes a complex manifold. Let be the set of all isomorphism classes of line bundles on X. . Where the second map is yields a homomorphism of groups: The image of a line bundle under this map is denoted by and is called the first Chern class of that is locally a finite sum.[1] The set of all divisors on X is denoted by . . -form. Such that.
Complex projective space. In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cn+1), Pn(C) or CPn. When n = 1, the complex projective space CP1 is the Riemann sphere, and when n = 2, CP2 is the complex projective plane (see there for a more elementary discussion). Complex projective space was first introduced by von Staudt (1860) as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. . .
Several complex variables. On the space Cn of n-tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct character, these are not just any functions: they are supposed to be holomorphic or complex analytic, so that locally speaking they are power series in the variables zi. Equivalently, as it turns out, they are locally uniform limits of polynomials; or locally square-integrable[citation needed] solutions to the n-dimensional Cauchy–Riemann equations. Historical perspective[edit] Many examples of such functions were familiar in nineteenth-century mathematics: abelian functions, theta functions, and some hypergeometric series.
Whenever n > 1. After 1945 important work in France, in the seminar of Henri Cartan, and Germany with Hans Grauert and Reinhold Remmert, quickly changed the picture of the theory. C.L. Subsequent developments included the hyperfunction theory, and the edge-of-the-wedge theorem, both of which had some inspiration from quantum field theory. The Cn space[edit] Runge's theorem.
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 it in the year 1885.
It states the following: are in A. Note that not every complex number in A needs to be a pole of every rational function of the sequence . We merely know that for all members of that do have poles, those poles lie in A. One aspect that makes this theorem so powerful is that one can choose the set A arbitrarily. For the special case in which C\K is a connected set (or equivalently that K is simply-connected), the set A in the theorem will clearly be empty. That approaches f uniformly on K. Proof[edit] An elementary proof, given in Sarason (1998), proceeds as follows. For w in K. See also[edit] Mergelyan's theorem References[edit] List of complex analysis topics. List of complex analysis topics From Wikipedia, the free encyclopedia Jump to: navigation, search Contents [hide] Overview[edit] Related fields[edit] Main article: Applied mathematics Local theory[edit] Growth and distribution of values[edit] Contour integrals[edit] Special functions[edit] Riemann surfaces[edit] Other[edit] Several complex variables[edit] History[edit] Main article: History of complex analysis People[edit] Retrieved from " Categories: Hidden categories: Navigation menu Personal tools Namespaces Variants Views Actions Navigation Interaction Tools Print/export Languages This page was last modified on 5 April 2013 at 12:02.
Complex dynamics. Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques[1][edit] Parts[edit] Holomorphic dynamics ( dynamics of holomorphic functions )[3]in one complex variablein several complex variablesConformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable.
See also[edit] References[edit] Complex dynamics, Lennart Carleson, Theodore W. Complex analysis. Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics". Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
History[edit] Complex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Important mathematicians associated with complex analysis include Euler, Gauss, Bernhard Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex functions[edit] For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts: and where are real-valued functions. In other words, the components of the function f(z), Holomorphic functions[edit] Major results[edit] Cauchy's integral formula. In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result denied in real analysis. Theorem[edit] where the contour integral is taken counter-clockwise. The proof of this statement uses the Cauchy integral theorem and similarly only requires f to be complex differentiable. Since the reciprocal of the denominator of the integrand in Cauchy's integral formula can be expanded as a power series in the variable (a − z0), it follows that holomorphic functions are analytic.
In particular f is actually infinitely differentiable, with where. Uniform convergence. The concept is important because several properties of the functions fn, such as continuity and Riemann integrability, are transferred to the limit f if the convergence is uniform. Uniform convergence to a function on a given interval can be defined in terms of the uniform norm. History[edit] Some historians claim[who?] That in 1821 Augustin Louis Cauchy published a false statement, but with a purported proof, that the pointwise limit of a series of continuous functions is always continuous; however, Lakatos offers a re-assessment[citation needed] of Cauchy's approach. The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series is independent of the variables and While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.[1] Notes[edit] with.
Uniform limit theorem. Counterexample to a strengthening of the uniform limit theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions converge to the non-continuous red function. This can happen only if convergence is not uniform. In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous. Statement[edit] More precisely, let X be a topological space, let Y be a metric space, and let ƒn : X → Y be a sequence of functions converging uniformly to a function ƒ : X → Y. This theorem does not hold if uniform convergence is replaced by pointwise convergence. In terms of function spaces, the uniform limit theorem says that the space C(X, Y) of all continuous functions from a topological space X to a metric space Y is a closed subset of YX under the uniform metric. The uniform limit theorem also holds if continuity is replaced by uniform continuity. Proof[edit] Let us fix an arbitrary ε > 0.