Definition, Betydelse & Synonymer | Engelska ordet HOLOMORPHIC

Definition av HOLOMORPHIC

1. (komplex analys) holomorf

Exempel på hur man kan använda HOLOMORPHIC i en mening

• A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane.
• In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane.
• In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space.
• In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities.
• Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them.
• These equations are in fact valid even for complex values of , because both sides are entire (that is, holomorphic on the whole complex plane) functions of , and two such functions that coincide on the real axis necessarily coincide everywhere.
• In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane.
• More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc.
• John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems.
• 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.
• Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: any pole must coincide with a zero of the denominator.
• In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
• 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.
• Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X.
• The Weierstrass preparation theorem implies that rings of germs of holomorphic functions are Noetherian rings.
• The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus.
• Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted as cohomology classes on the associated Riemann surfaces via the Eichler-Shimura isomorphism.
• Subsequent generalizations extended Nevanlinna theory to algebroid functions, holomorphic curves, holomorphic maps between complex manifolds of arbitrary dimension, quasiregular maps and minimal surfaces.
• A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions.
• The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it.

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