In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.
Let and be Riemann surfaces, and let be a non-constant holomorphic map. Fix a point and set . Then there exist and charts on and on such that
This theorem gives rise to several definitions:
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
In complex analysis, the Riemann mapping theorem states that if is a non-empty simply connected open subset of the complex number plane which is not all of , then there exists a biholomorphic mapping from onto the open unit disk
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism.
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.
A covering of a topological space is a continuous map with special properties.
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables, which the Mathematics Subject Classification has as a top-level heading.
In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that if the function is n-valued at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951). A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
The following are important identities involving derivatives and integrals in vector calculus.
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. It is illustrated in the figure, where the direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule. For the right hand the fingers circulate along ∂Σ and the thumb is directed along n.
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity. With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers.
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation
In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space. Within the language of potential theory it reduces the partial differential equation to an integral equation on the boundary to which the theory of Fredholm operators can be applied. The theory is particularly simple in two dimensions—the case treated in detail in this article—where it is related to complex function theory, the conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains.
In mathematics, a planar Riemann surface is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910, as a generalization of the uniformization theorem, that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.
In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later Weyl (1940) found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of Hodge (1941). This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.
In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somehow more sophisticated in that it uses linear algebra more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.
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