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In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood...

the study of holomorphic functions that are the differentiable functions of a complex variable. By contrast with the real case, a holomorphic function is...

π : E → X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle...

mathematics, in the field of complex geometry, a holomorphic curve in a complex manifold M is a non-constant holomorphic map f from the complex plane to M. Nevanlinna...

{\displaystyle \mathbb {C} ^{n}} , such that the transition maps are holomorphic. The term "complex manifold" is variously used to mean a complex manifold...

complex analysis of functions of one variable the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables...

in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or...

a complex function on an open set is analytic if and only if it is holomorphic, that is, complex differentiable at every point of the set. For this...

line integrals for holomorphic functions in the complex plane. Essentially, it says that if f ( z ) {\displaystyle f(z)} is holomorphic in a simply connected...

central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on...

{\displaystyle f} of a complex variable z {\displaystyle z} : is said to be holomorphic at a point a {\displaystyle a} if it is differentiable at every point...

biholomorphic mapping f {\displaystyle f} (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U {\displaystyle U} onto the open unit disk...

algebraic geometry, a formal holomorphic function along a subvariety V of an algebraic variety W is an algebraic analog of a holomorphic function defined in a...

(squared) pointwise norm | ∂ f | 2 {\displaystyle |\partial f|^{2}} of a holomorphic map f : ( X , g X ) → ( Y , g Y ) {\displaystyle f:(X,g_{X})\to (Y,g_{Y})}...

after Giacinto Morera, gives a criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function f...

functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable z {\displaystyle z} defined...

subset D {\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...

In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a...

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...

{\textstyle \lambda } , and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This...

especially complex geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of the tangent bundle...

{C} ^{n}} ) has nonpositive holomorphic sectional curvature. For holomorphic maps between Hermitian manifolds, the holomorphic sectional curvature is not...

Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K {\displaystyle K} with closed contour ∂ K {\displaystyle...

Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that the support...

that are holomorphic functions of τ. A quasimodular form is the holomorphic part of an almost holomorphic modular form. An almost holomorphic modular form...

principle in complex analysis states that if f {\displaystyle f} is a holomorphic function, then the modulus | f | {\displaystyle |f|} cannot exhibit a...

varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of...

function in a normal family F satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set F. More...

that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number...

biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formally, a biholomorphic function is a function...

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 z0. A function...

of so-called (p, q)-forms: roughly, wedges of p differentials of the holomorphic coordinates with q differentials of their complex conjugates. The ensemble...

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...

In mathematics, a weakly holomorphic modular form is similar to a holomorphic modular form, except that it is allowed to have poles at cusps. Examples...

space of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian. Holomorphic discrete...

n-space is holomorphic if it is holomorphic in each complex coordinate separately. Several complex variables is the study of such holomorphic functions...

functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces...

dictionary. Entire may refer to: Entire function, a function that is holomorphic on the whole complex plane Entire (animal), an indication that an animal...

In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form...

hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as...

harmonic functions of two variables are: The real or imaginary part of any holomorphic function. In fact, all harmonic functions defined on the plane are of...

specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map, from a Riemann surface into an almost complex...

{\displaystyle f:U\to \mathbb {C} } is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on U {\displaystyle U} . If...

} Holomorphic Fock space (also known as the Segal–Bargmann space) is defined to be the vector space F {\displaystyle {\mathcal {F}}} of holomorphic functions...

point at infinity). A meromorphic function is a complex function that is holomorphic and therefore analytic everywhere in its domain except at a finite, or...

families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic functions...

of holomorphic functions on C n {\displaystyle \mathbb {C} ^{n}} (and subsequently the sheaf O X {\displaystyle {\mathcal {O}}_{X}} of holomorphic functions...

work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook...

U_{0}=U\smallsetminus \{a_{1},\ldots ,a_{n}\}} ⁠, and a function f {\displaystyle f} holomorphic on ⁠ U 0 {\displaystyle U_{0}} ⁠. Letting γ {\displaystyle \gamma } be...

can be taken more narrowly, as some Banach space, often consisting of holomorphic functions: for example, some Hardy space or Bergman space. In this case...

The Holomorphic Embedding Load-flow Method (HELM)  is a solution method for the power-flow equations of electrical power systems. Its main features are...

the transition maps between two overlapping charts are required to be holomorphic. A Riemann surface is a (connected) oriented manifold of (real) dimension...

analysis, Cauchy's estimate gives local bounds for the derivatives of a holomorphic function. These bounds are optimal. Cauchy's estimate is also called...

{fk} be a sequence of holomorphic functions on a connected open set G that converge uniformly on compact subsets of G to a holomorphic function f which is...

{C} \smallsetminus \{a_{k}\}_{k}\rightarrow \mathbb {C} } ⁠ that is holomorphic except at the discrete points ⁠ { a k } k {\displaystyle \{a_{k}\}_{k}}...

function, as defined above for real positive s and x, can be developed into holomorphic functions, with respect both to x and s, defined for almost all combinations...

and f : U → C {\displaystyle f:U\to \mathbb {C} } is a non-constant holomorphic function, then f {\displaystyle f} is an open map (i.e. it sends open...

function. There are also versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable...

stability problems. The classical Paley–Wiener theorems make use of the holomorphic Fourier transform on classes of square-integrable functions supported...

corresponding holomorphic vector bundle with fibre A is the tangent bundle of the complex manifold X. Its holomorphic sections are just holomorphic vector fields...

In mathematics, the Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the...

about the behavior of holomorphic functions. Hadamard three-circle theorem: Let f ( z ) {\displaystyle f(z)} be a holomorphic function on the annulus...

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective...

sheaf of rings O X {\displaystyle {\mathcal {O}}_{X}} , the sheaf of holomorphic functions or regular functions, and coherent sheaves are defined as a...

In mathematical physics, six-dimensional holomorphic Chern–Simons theory or sometimes holomorphic Chern–Simons theory is a gauge theory on a three-dimensional...

{\displaystyle \mathbb {C} \cup \{\infty \}} and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex...

Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by iterating a complex analytic mapping. This article focuses on...

Kobayashi hyperbolicity of a complex manifold X implies that every holomorphic map from the complex line C to X is constant. The origins of the concept...

describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold. If v is a holomorphic vector field on a compact complex manifold...

related. He founded, independently with Pierre Fatou, the modern theory of holomorphic dynamics. Julia was born in the Algerian town of Sidi Bel Abbès, at the...

{\displaystyle \operatorname {Log} z} at a negative real number. Like every holomorphic function, the complex logarithm can be represented locally – near any...

Contour integration is used to study complex-valued functions that are holomorphic in a region. Contour integration is closely related to the calculus of...

u={\text{const}}} curves are the equipotential curves of the flow. A holomorphic function can therefore be visualized by plotting the two families of...

itself. In particular, the Oka coherence theorem states that the sheaf of holomorphic functions on a complex analytic space X {\displaystyle X} is a coherent...

{C} } is holomorphic. If f {\displaystyle f} is holomorphic at all x ∈ X {\displaystyle x\in X} , we say f {\displaystyle f} is holomorphic. The map F...

symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space...

which may have nodes or (a finite tree of) "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve...

intrinsically it can be defined as a complex manifold admitting a proper holomorphic embedding into C n {\displaystyle \mathbb {C} ^{n}} for some n {\displaystyle...

example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function...

that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician...

{\displaystyle z\geq 0} is operator monotone if and only if it extends to a holomorphic function on C ∖ ( − ∞ , 0 ] {\displaystyle \mathbb {C} \setminus (-\infty...

Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex...

Narasimhan–Seshadri theorem, proved by Narasimhan and Seshadri (1965), says that a holomorphic vector bundle over a compact Riemann surface is stable if and only if...

solutions to Hitchin's equations give examples of Higgs bundles and of holomorphic connections. The existence of solutions to Hitchin's equations on a compact...

is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does not take the values 0 or 1...

fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series. The connectedness assumption on the...

compact space is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise...

In mathematics, the edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

holomorphic functional calculus is defined as follows: Fix a bounded operator T. Consider the family Hol(T) of complex functions that is holomorphic on...

In mathematics, the Milne-Thomson method is a method for finding a holomorphic function whose real or imaginary part is given. It is named after Louis...

variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. The Oka–Weil...

reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic functions on a domain D in Cn. In detail, let L2(D) be the Hilbert space...

It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had...

analytic continuation An analytic continuation of a holomorphic function is a unique holomorphic extension of the function (on a connected open subset...

behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists...

statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. If we have a holomorphic function f {\displaystyle...

real affine functions or hyperplanes, it uses inequalities involving holomorphic polynomials. Let K {\displaystyle K} be a compact subset of C n {\displaystyle...

or space of holomorphic sections is larger than expected. Alternatively, by Serre duality, the condition is that there exist holomorphic differentials...

(E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle \varphi } , a holomorphic 1-form taking values in the bundle...

write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves...

two points of a holomorphic function as the sum of the products of the coordinate differences of these two points with other holomorphic functions defined...

result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have...

_{i}^{j}\wedge \omega ^{i}} for some system of holomorphic 1-forms ψ j i, 1 ≤ i, j ≤ r, then there exist holomorphic functions fij and gi such that, on a possibly...

known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions F in n complex variables satisfying the square-integrability...

atlas. This chain can be extended to include holomorphic atlases, with the understanding that any holomorphic map between open subsets of Cn can be viewed...

function Holomorphic function Meromorphic function Cauchy–Riemann equations Formal power series Laurent series Theorems Analyticity of holomorphic functions...

required to be holomorphic. The latter condition allows one to transfer the notions and methods of complex analysis dealing with holomorphic and meromorphic...

sheaf of holomorphic functions and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function...

manifold. For an appropriate choice of almost complex structure, punctured holomorphic curves (of finite energy) in it have cylindrical ends asymptotic to the...

mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor...

exist even for holomorphic functions of a complex variable. For example, consider the reciprocal function, g(z) = 1/z which is holomorphic on the punctured...

a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved...

differentiable on an open subset of C {\displaystyle \mathbb {C} } is holomorphic and hence analytic on that set. Differentiability class is a classification...

distinguished class of projective representations of loop groups. They extend holomorphically to the complexified loop group and decompose into finite-dimensional...

first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global...

manifold ( M , I ) {\displaystyle (M,I)} , is holomorphically symplectic (equipped with a holomorphic, non-degenerate, closed 2-form). More precisely...

adapted to the complex case. A holomorphic Poisson manifold is a complex manifold M {\displaystyle M} whose sheaf of holomorphic functions O M {\displaystyle...

{\displaystyle {\bar {z}}} . These are referred to as the holomorphic and anti-holomorphic parts respectively, as they are continuous and differentiable...

Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf...

a 2-form, but is in fact a (1, 1)-form (see holomorphic vector bundle#Hermitian metrics on a holomorphic vector bundle). Hence, the Chern–Weil homomorphism...

connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's...

In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. The purpose of the theory is to provide...

sphere packing, and string theory. More precisely, a modular form is a holomorphic function on the complex upper half-plane that roughly satisfies a functional...

Open mapping theorem (complex analysis), states that a non-constant holomorphic function on a connected open set in the complex plane is an open mapping...

pairs of functions F+ and F− such that F+ is holomorphic on the upper half-plane and F− is holomorphic on the lower half-plane, such that for x along...

forms is still an open problem, as the Deligne method which solves the holomorphic case does not work in the real-analytic case of Maass forms. The generalization...

complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold...

to reprove theorems of Riemann. In modern language, if ω1 and ω2 are holomorphic differentials on an algebraic curve C, then their wedge product is necessarily...

\mathbb {D} } , the Nevanlinna–Pick interpolation problem is to find a holomorphic function φ {\displaystyle \varphi } that interpolates the data, that...

contour integral formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions...

employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function f {\displaystyle f} (i.e., | f ( z ) | < M     ( z ∈ Ω ) {\displaystyle...

"Cauchy–Riemann" or "Complex-Real". A CR-function, which is a generalization of a holomorphic function, is a solution to the system of equations ∂ b ¯ f = 0 {\displaystyle...

holomorphy is a domain that is maximal in the sense that there exists a holomorphic function on this domain that cannot be extended to a bigger domain. Formally...

scalar fields. The dynamics of the chiral multiplets is fixed by the holomorphic superpotential W ( ϕ ) {\displaystyle W(\phi )} and the Kähler potential...

bundle E {\displaystyle E} on X {\displaystyle X} is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part...

Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first...

dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch...

ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions...

Bloch and denoted B {\displaystyle {\mathcal {B}}} or ℬ, is the space of holomorphic functions f defined on the open unit disc D in the complex plane, such...

German mathematician Hermann Schwarz. The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by ( S f ) ( z ) = (...

^{n}} allowing holomorphic extension in each variable separately can be extended, under certain conditions, to a function holomorphic in all variables...

continuous functional calculus, and its focus is different than the holomorphic functional calculus. More precisely, the Borel functional calculus allows...

the calculations in topological string theory generically encode all holomorphic quantities within the full string theory whose values are protected by...

results such as the Ahlfors's Five Islands theorem, Cartan's theorem on holomorphic curves omitting hyperplanes, Hayman's result that an exceptional set...

g (resp. h) is a holomorphic (resp. antiholomorphic) function on D. When one also asks for the harmonic extension to be holomorphic, then the solutions...

Complex numbers Complex plane Complex functions Complex derivative Holomorphic functions Harmonic functions Elementary functions Polynomial functions...

Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle...

dynamics, the connectedness locus of a parameterized family of one-variable holomorphic functions is a subset of the parameter space which consists of those...

y ) ↦ x y − 1 {\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}} is holomorphic. Basic examples are GL n ⁡ ( C ) {\displaystyle \operatorname {GL} _{n}(\mathbb...

mathematics, the modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional...

characterised by being the only simply connected complete Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian...

groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from...

in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R), with...

Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects...

the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex...

locally be written as f(z)/(z − z0)n with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential...

) {\displaystyle (E,\Phi )} where E → X {\displaystyle E\to X} is a holomorphic vector bundle and Φ : E → E ⊗ Ω 1 {\displaystyle \Phi :E\to E\otimes...

{\displaystyle D} ⁠ centered at ⁠ z 0 {\displaystyle z_{0}} ⁠ such that f is holomorphic on ⁠ D ∖ { z 0 } {\displaystyle D\smallsetminus \{z_{0}\}} ⁠, that is...

describes the remarkable behaviour of holomorphic functions near essential singularities, which is that every holomorphic function gets values from any complex...

\delta ;z)} is the solution of Heun's differential equation that is holomorphic and 1 at the singular point z = 0. The local Heun function is called...

a branch of mathematics, the residue at infinity is a residue of a holomorphic function on an annulus having an infinite external radius. The infinity...

theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of the unit disk in the complex numbers that fixes...

{\displaystyle f} is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions...

of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie...

depending on the function's zeros and poles, and an associated non-zero holomorphic function.[citation needed] It is clear that any finite set { c n } {\displaystyle...

over a Riemann surface decomposes into a space that is holomorphic, and one that is anti-holomorphic: Ω 1 = Ω 1 , 0 ⊕ Ω 0 , 1 {\displaystyle \Omega ^{1}=\Omega...

function Holomorphic function Meromorphic function Cauchy–Riemann equations Formal power series Laurent series Theorems Analyticity of holomorphic functions...

Hardy spaces (or Hardy classes) H p {\displaystyle H^{p}} are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced...

complex analysis describing the behavior of holomorphic functions. Let f {\displaystyle f} be a holomorphic function on the open ball centered at zero...

spherical functions on a semisimple Lie group, and a certain space of holomorphic functions defined on a tubular neighborhood of the dual of a Cartan subalgebra...

questions. It also indicates a connection between the combinatorics of J-holomorphic curves in the blow up of the projective plane and the numbers that appear...

characterize holomorphic functions on a domain in the complex plane have an analogue for functions of a motor variable. An approach to D-holomorphic functions...

determinant bundle of the holomorphic cotangent bundle T ∗ V {\displaystyle T^{*}V} . Equivalently, it is the line bundle of holomorphic n {\displaystyle n}...

the supervision of Isadore Singer in 1960 (Maximality of algebras of holomorphic functions). In 1960 he became an assistant professor at the University...

depends only on the ratio ⁠r/R⁠. Each annulus ann(a; r, R) can be holomorphically mapped to a standard one centered at the origin and with outer radius...

'local holomorphic coordinates for J'. If M admits local holomorphic coordinates for J around every point then these patch together to form a holomorphic atlas...

Hadamard's gamma function plotted over part of the real axis. Unlike the classical gamma function, it is holomorphic; there are no poles....

is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore...

real-valued functions on U, differential forms on U, vector fields on U, holomorphic functions on U (when X is a complex manifold), constant functions on...

isomorphism classes of topological and holomorphic vector bundles on affine space, so there are no non-trivial holomorphic vector bundles either. Jean-Pierre...

function Holomorphic function Meromorphic function Cauchy–Riemann equations Formal power series Laurent series Theorems Analyticity of holomorphic functions...

continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles....

analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself. The result...

Stefan Bergman Prize. Berndtsson's first results concern zero sets of holomorphic functions, and in 1981 he showed that any divisor with finite area in...

to recover a holomorphic function, up to an imaginary constant, from the boundary values of its real part. Let f be a function holomorphic on the closed...

holomorphic functions on Ω {\displaystyle \Omega } , with the topology of uniform convergence on compact subsets. Many Banach spaces of holomorphic functions...

Prize. C. Camacho, P. Sad. "Invariant varieties through singularities of holomorphic vector fields", Annals of Mathematics, 1982 C. Camacho, A. L. Neto, P...

to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. The theorem is named after the Swedish...

Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition...

justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the spectral properties of...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

is a bijection between instantons on complexified Minkowski space and holomorphic vector bundles on twistor space, which as a complex manifold is P 3 {\displaystyle...

importance in complex analysis, where they are intimately connected to holomorphic functions. One can show that a real-valued, continuous function φ {\displaystyle...

generalizations extended Nevanlinna theory to algebroid functions, holomorphic curves, holomorphic maps between complex manifolds of arbitrary dimension, quasiregular...

\mathbb {C} } and f : Ω → C {\displaystyle f:\Omega \to \mathbb {C} } a holomorphic function. If f {\displaystyle f} is not constant, then the set of the...

Seshadri proved the Narasimhan–Seshadri theorem, which relates stable holomorphic (or algebraic) vector bundles over compact Riemann surfaces (or non-singular...

if x = y. The prime form E is not quite a holomorphic function on X × X, but is a section of a holomorphic line bundle over this space. Prime forms were...

{\displaystyle \gamma } of Γ {\displaystyle \Gamma } acting by holomorphic contractions. Here, a holomorphic contraction is a map γ : C n → C n {\displaystyle \gamma...

Superiore di Pisa. His dissertation thesis was titled Iteration Theory of Holomorphic Maps on Taut Manifolds. His doctoral advisor was Edoardo Vesentini. He...

power of the prime p Modular lambda function, λ(τ), a highly symmetric holomorphic function on the complex upper half-plane Carmichael function, λ(n), in...

of the Kobayashi–Hitchin correspondence for principal bundles, that a holomorphic principal bundle over a compact Kähler manifold admits a Hermite–Einstein...

({\bar {\partial }}:\Omega ^{p,q-1}\to \Omega ^{p,q})}}.} If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a...

under Halsey Royden. His Ph.D. thesis was titled: Holomorphic Mappings and algebras of holomorphic functions of several complex variables. He has previously...

appears to be a well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective...

model space is an open subset of the vanishing locus of a finite set of holomorphic functions. Complex analytic varieties are analogous to algebraic varieties...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

born in Nice, France and died in Paris, France. He researched mostly on holomorphic functions in complex analysis. Montel was a student of Émile Borel at...

University under Phillip Griffiths with thesis Some Picard Theorems for Holomorphic Maps to Algebraic Varieties. In 1970/71, Green was a Procter Fellow in...

theorem is a fundamental result concerning the holomorphic extension of an L 2 {\displaystyle L^{2}} -holomorphic function defined on a bounded Stein manifold...

standard holomorphic function S → J ( S ) {\displaystyle S\to J(S)} of complex manifolds. It has the defining property that the holomorphic 1-forms on...

several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays...

former chair of the Danish Mathematical Society. Her research concerned holomorphic dynamics and the history of mathematics. Branner studied mathematics...

In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface. It is a generalization...

geometry of projective complex varieties. For example, the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide...

Convex hull, in convex geometry Carathéodory's theorem (convex hull) Holomorphically convex hull, in complex analysis Injective hull, of a module Linear...

also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are...

analysis studies holomorphic functions, the differentiable functions of a complex variable. By contrast with the real case, a holomorphic function is always...

complex formulae. The above theorem generalizes in the obvious way to holomorphic functions: Let U {\displaystyle U} and V {\displaystyle V} be two open...

Fields Mathematics Institutions Cornell University Thesis Construction of Holomorphic Differential Forms on the Moduli Space of Abelian Varieties  (1983) Doctoral...

holomorphic contact structure is apparent, because the nilpotent orbits of semisimple Lie groups are equipped with the Kirillov-Kostant holomorphic symplectic...

n ) n = 1 ∞ {\displaystyle (f_{n})_{n=1}^{\infty }} is a sequence of holomorphic functions f n : Ω → C {\displaystyle f_{n}:\Omega \to \mathbb {C} } that...

continuous function of several complex variables that is holomorphic in each variable separately is holomorphic. The assumption that the function is continuous...

two-dimensional planes which are holomorphic, i.e. which are invariant under the almost-complex structure. This is called the holomorphic sectional curvature. Samuel...

sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C {\displaystyle \mathbb {C} } can...

a Kähler manifold except the requirement that the transition maps be holomorphic. Gromov used the existence of almost complex structures on symplectic...

holomorphic functions f in D that are continuous on the closure of D. As a result, the delta function δz is represented in this class of holomorphic functions...

In complex analysis, the Schur class is the set of holomorphic functions f ( z ) {\displaystyle f(z)} defined on the open unit disk D = { z ∈ C : | z...

Szabó for: Holomorphic disks and topological invariants for closed three-manifolds. Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. Holomorphic disks and...

Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane...

H^{1}(\Theta )\,} where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle. There is an obstruction in the H2 of the same sheaf;...

macromolecules, a method of describing complex biological molecules Holomorphic embedding load flow method, a mathematical technique for solving AC power...

1)-classes, named after Solomon Lefschetz, is a classical statement relating holomorphic line bundles on a compact Kähler manifold to classes in its integral...

topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex...

the Jordan normal form of A; that is, A = C−1JC. Now let f (z) be a holomorphic function on an open set Ω {\displaystyle \Omega } such that s p e c A...

universal function is one that contains subregions that approximate every holomorphic function to arbitrary accuracy. The Riemann zeta function (and some others)...

 n=0 ). The first summand is called the holomorphic part, and the second summand is called the non-holomorphic part of f . {\displaystyle f.} There is...

University, US Harvard University, US "For his contributions to the theory of holomorphic dynamics and geometrization of three-manifolds, including proofs of Bers'...

f {\displaystyle f} to int ( K ) {\displaystyle {\text{int}}(K)} is holomorphic, can be approximated uniformly on K {\displaystyle K} with polynomials...

ISBN 978-3-540-63005-0. Vitushkin, A. G. (1988). "Uniform approximation of functions by holomorphic functions". Proc. Steklov Inst. Math. (Mathematical Physics and Complex...

graded vector space V(g), and to each commuting pair of elements (g, h) a holomorphic function f(g, h, τ) on the upper half-plane, such that: Each V(g) is...

led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed...

Stein 1970-02-21 160 9780691080673 64 The Equidistribution Theory of Holomorphic Curves Hung-Hsi Wu 1970-02-21 250 978-0691080734 66 Advances in the Theory...

as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The universality of the Riemann zeta function...

pioneering contributions to the theory of iteration and early developments in holomorphic dynamics, including Böttcher's equation and Böttcher's theorem. He worked...

In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach...

and only if they are respectively the real and imaginary parts of a holomorphic function f ( z ) {\displaystyle f(z)} of the complex variable z := x...

extreme values The Weierstrass–Casorati theorem describes the behavior of holomorphic functions near essential singularities The Weierstrass preparation theorem...

instances of the Oka principle on deformation of continuous maps to holomorphic maps.[G89] His work initiated a renewed study of the Oka–Grauert theory...

occurrences, such as in the 24-dimensional Leech lattice. The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it...

is a removable singularity of f {\displaystyle f} if there exists a holomorphic function g {\displaystyle g} defined on all of U {\displaystyle U} such...

analysis: Let f(z) be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated...

hyperbolic surfaces, 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic...

{\displaystyle E\to \Sigma } is a holomorphic vector bundle and Φ : E → E ⊗ K {\displaystyle \Phi :E\to E\otimes K} is a holomorphic endomorphism of E {\displaystyle...

Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions. List of transforms List of Fourier-related transforms Transfer...

the basic results of holomorphic dynamics were also independently obtained by Gaston Julia and Samuel Lattes in 1918. Holomorphic dynamics has experienced...

T^{*}(y_{n_{k}}^{*})} of that subsequence is Cauchy in X∗. When f is holomorphic in an open disk D1 = B(z0, r), with modulus bounded by M, then (for example...

Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions. The first proof of Carathéodory's theorem presented here is...

after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

equations. Hodge–Arakelov theory Holomorphic functional calculus a branch of functional calculus starting with holomorphic functions. Homological algebra...

half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that j ( e 2 π i / 3 ) = 0 ...

879–897. Kim, Kang-Tae; Poletsky, Evgeny; Schmalz, Gerd Functions holomorphic along holomorphic vector fields. J. Geom. Anal. 19 (2009), no. 3, 655–666. 32A10...

∂ n } {\displaystyle \{\partial _{1},\ldots ,\partial _{n}\}} of the holomorphic tangent bundle of CPn, in terms of which the Fubini–Study metric has...

invariant theory, with the idea of studying holomorphic vector fields along projective embeddings, rather than holomorphic vector fields on the manifold itself...

with successive convergents of a continued fraction representation of a holomorphic or meromorphic function. Although earlier mathematicians had obtained...

function Holomorphic function Meromorphic function Cauchy–Riemann equations Formal power series Laurent series Theorems Analyticity of holomorphic functions...

Hermitian metric on any holomorphic vector bundle over X {\displaystyle X} (note that the Levi-Civita connection on the holomorphic tangent bundle is precisely...

\left\{z\in \mathbb {C} \mid \operatorname {Im} (z)\geq 0\right\}} , holomorphic on the upper half plane { z ∈ C ∣ Im ⁡ ( z ) > 0 } {\displaystyle \left\{z\in...

difference between C*-algebras and general Banach algebras, in which only a holomorphic functional calculus exists. If one wants to extend the natural functional...

following statement: Let (M, g) be a closed Kähler manifold of nonnegative holomorphic bisectional curvature. Then the universal cover of M, with its natural...

surface has constant mean curvature, then their Hopf differential is holomorphic relative to the natural complex structure on the surface. As an application...

integers n) and compactly supported if and only if f̂ (σ + iτ) is a holomorphic function for which there exists a constant a > 0 such that for any integer...

L_{d})} of the cochain complexes, the moduli space of J {\displaystyle J} -holomorphic polygons with d + 1 {\displaystyle d+1} faces with each face mapped into...

that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Such a function is necessarily infinitely differentiable...

"On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations", Mathematische Zeitschrift, 276 (1–2): 505–515, arXiv:1207...

_{k}=\varepsilon _{1}+\cdots +\varepsilon _{k}} . Equivalently, in the holomorphic Fock model, the two components may be viewed, depending on the choice...

equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either C {\displaystyle \mathbb {C} } or to the...

collaboration with James Newton, of the automorphy of all symmetric powers of a holomorphic modular newform." Jacob Tsimerman – "For outstanding work in analytic...

domain where its derivative is equal to zero (or the function is not holomorphic). Likewise, for a function of several real variables, a critical point...

operator is actually bounded. Using the tools of holomorphic functional calculus, given a holomorphic function f {\displaystyle f} defined on an open set...

(D)=\chi (0)+{\tfrac {1}{2}}D.(D-K)\,} where χ {\displaystyle \chi } is the holomorphic Euler characteristic, the dot . {\displaystyle .} is the intersection...

dissertation, Leau examined, among other things, the iteration behavior of holomorphic functions in the environment of a rationally indifferent fixed point...

series. In this context the extension of holomorphic functions of a complex variable is developed as the holomorphic functional calculus. Hypercomplex analysis...

property of the graph of a function near a zero Zeros and poles of holomorphic functions Foerster, Paul A. (2006). Algebra and Trigonometry: Functions...

meromorphic function f. The meromorphic differential df = dU + idV is holomorphic everywhere except for a double pole at P with singular term d(z−1) at...

Many operators that are studied are operators on Hilbert spaces of holomorphic functions, and the study of the operator is intimately linked to questions...

{\displaystyle C_{p}} such that for every p {\displaystyle p} -valent holomorphic function f ( z ) = ∑ i = 0 ∞ a n z n {\displaystyle f(z)=\sum _{i=0}^{\infty...

complex-valued function that is holomorphic everywhere, apart from at isolated points where there are poles. Entire function: A holomorphic function whose domain...

in these operators. Holomorphic functional calculus, which attempts to extend the techniques commonly used to study holomorphic functions f ( z ) {\displaystyle...

sheaf of real-analytic functions on M is the restriction of the sheaf of holomorphic functions on X to M. Hyperfunction D-module Microlocal analysis Generalized...

coordinate transitions between two different such charts Ui and Uj are holomorphic functions (in fact they are fractional linear transformations). Thus...

functions generalizes as follows for holomorphic functions: near a non-degenerate saddle point z0 of a holomorphic function S(z), there exist coordinates...

( S ) {\displaystyle f\in \operatorname {Diff} (S)} such that: It is holomorphic (the differential is complex linear at each point, for the structures...

calculate the integral along the real axis of functions f(z) = ei a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane...

operator which maps modular forms to almost holomorphic modular forms. The Maass–Shimura operator on (almost holomorphic) modular forms of weight k {\displaystyle...

endomorph, endomorphic, epimorphism, geomorphology, hemimorphic, holomorph, holomorphic, holomorphism, homeomorphic, homeomorphism, homomorphic, homomorphism...

function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred...

In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states...

Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic (really, meromorphic) properties and their behavior near the singularities...

_{\mathbb {P} ^{n}}.} The Nullstellensatz also holds for the germs of holomorphic functions at a point of complex n-space C n . {\displaystyle \mathbb...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

share some of the basic properties with holomorphic functions of one complex variable. The theory of holomorphic (=analytic) functions of one complex variable...

The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement...

H} are the restrictions to R {\displaystyle \mathbb {R} } of entire holomorphic functions, by the Paley–Wiener theorem. From the Fourier inversion theorem...

a holomorphic vector bundle with a Hermitian structure, is the unique metric connection D, such that the part which increases the anti-holomorphic type...

Newton, Thorne has established symmetric power functoriality for all holomorphic modular forms. Thorne was awarded the Whitehead Prize in 2017. In 2018...

treated teleomorphic names as having precedence for being used as the holomorphic name. The Melbourne Code removes the provisions and allows all names...

Specifically, a Riemann–Hilbert problem is a boundary value problem for a holomorphic function on the complement of an oriented contour in the complex plane...

only if the Lie bracket of two sections of the holomorphic subbundle is another section of the holomorphic subbundle. In generalized complex geometry one...

{f_{r}(z))=\sum _{n\geq 0}r^{n}a_{n}z^{n}}} is holomorphic on |z| < 1/r. In that case fr(T) is defined by the holomorphic functional calculus and f (T ) can be...

mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula...

Potier (1975): Let X be a n-dimensional compact complex manifold and E a holomorphic vector bundle of rank r over X, here H p , q ( X , E ) {\displaystyle...

doi:10.1090/s0002-9904-1967-11758-0. MR 0210934. Rudin, Walter (1981). "Holomorphic maps that extend to automorphisms of a ball" (PDF). Proc. Amer. Math...

"Operator Theory in Function Spaces", "Theory of Bergman Spaces", "Spaces of Holomorphic Function Spaces on the Unit Ball", and "Analysis on Fock spaces". He...

^{n}\times \mathbf {P} ^{n-1}\to \mathbf {C} ^{n}} naturally induces a holomorphic map π : C n ~ → C n . {\displaystyle \pi :{\tilde {\mathbf {C} ^{n}}}\to...

{\displaystyle \Gamma } . In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949. The group G := S L 2...

particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X {\displaystyle X} is a complex manifold and if the...

Kriwet, Jürgen (2025-01-23). "Insights into stem Batomorphii: A new holomorphic ray (Chondrichthyes, Elasmobranchii) from the upper Jurassic of Germany"...

analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ : D → C {\displaystyle \mathbb {C} } (where D is the open...

Prize for Science Awardee. Pioneer in the study of moduli spaces of holomorphic vector bundles on projective varieties P. T. Narasimhan pioneer of computational...

than the upper half-plane by discrete groups. Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite...

equivalently, such that the product f g 2 {\displaystyle fg^{2}} is holomorphic), and let c 1 , c 2 , c 3 {\displaystyle c_{1},c_{2},c_{3}} be constants...

polynomial Barbeau 2003, pp. 1–2 Kriete, Hartje (1998-05-20). Progress in Holomorphic Dynamics. CRC Press. p. 159. ISBN 978-0-582-32388-9. This class of endomorphisms...

analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be...

understood, especially following work of Akito Futaki, that the existence of holomorphic vector fields should act as an obstruction to the existence of Kähler-Einstein...

its entire domain. This follows directly from the identity theorem for holomorphic functions. A common way to define functions in complex analysis proceeds...

part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of g. The...

conjugate V of U is the unique real function on Ω such that U + iV is holomorphic. As such it must satisfy the Cauchy–Riemann equations: U x = − V y ,...

algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves". Geometry & Topology. 20 (2): 779–1034. arXiv:1309.2370. doi:10...

Kähler–Einstein metric, and, more importantly, comes equipped with a holomorphic contact structure, corresponding to the horizontal spaces of the Riemannian...

Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in C n {\displaystyle \mathbb {C} ^{n}} can be extended...

Hungarian Academy of Sciences.[citation needed] ———; Ozsváth, Peter (2004), "Holomorphic disks and topological invariants for closed three-manifolds", Annals...

of California, Berkeley in 1995 under Curtis McMullen with the thesis Holomorphic removability of quadratic polynomial Julia sets. Kahn was assistant professor...

role of connections, Princeton University Press 1978 Introduction to holomorphic functions of several variables. 3 vols., Wadsworth and Brooks/Cole, 1990...

were not initially known. Physically, the vertex operators arising from holomorphic field insertions at points in two-dimensional conformal field theory...

against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon...

{f(it)-f(-it)}{e^{2\pi t}-1}}\,dt.} It holds for functions ƒ that are holomorphic in the region Re(z) ≥ 0, and satisfy a suitable growth condition in this...

MR 0176482. Zbl 0127.13401. Gromov, Mikhaïl (2003). "On the entropy of holomorphic maps" (PDF). L'Enseignement Mathématique. Revue Internationale. 2e Série...

American Mathematical Society. Ozsváth, Peter; Szabó, Zoltán (2004). "Holomorphic disks and topological invariants for closed three-manifolds". Ann. of...

proved that a ball in C 2 {\displaystyle \mathbb {C} ^{2}} is not holomorphically equivalent to the bidisc. He is credited with the Carathéodory extension...

power series with a positive radius of convergence can be made into a holomorphic function by taking its argument to be a complex variable. The radius...

on Riemann surfaces, and automorphe functions) Röhrl, Helmut (1962). "Holomorphic fiber bundles over Riemann surfaces". Bulletin of the AMS. 68 (3): 125–160...

continuous on this set of matrices. These properties are consequences of the holomorphic functional calculus applied to matrices. The existence and uniqueness...

Newton, James; Thorne, Jack (2021). "Symmetric power functoriality for holomorphic modular forms". Publications Mathématiques de l'IHÉS. 134: 1–116. doi:10...

Other contributions, not discussed here, include the construction of a holomorphic version of the long line with Maxwell Rosenlicht, a study of the moduli...

relationships and palaeobiogeographic implications of the first Neogene holomorphic stingray (Myliobatiformes: Dasyatidae) from the early Miocene of Sulawesi...

defined as follows: K is the canonical line bundle whose sections are the holomorphic 2-forms. P n = dim ⁡ H 0 ( K n ) , n ⩾ 1 {\displaystyle P_{n}=\dim H^{0}(K^{n})...

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function...

every homomorphism induces a holomorphic map, but every holomorphic map is the composition of a special kind of holomorphic map with a homomorphism. For...

f ( y ) ) | f : B → Δ  is holomorphic } . {\displaystyle d(x,y)=\sup\{\rho (f(x),f(y))|f:B\to \Delta {\mbox{ is holomorphic}}\}.} What it means for a...

{\displaystyle b} ) of a holomorphic function f on the domain D {\displaystyle D} is a function S {\displaystyle S} , holomorphic on domain D {\displaystyle...

matrix group to have a real analytic structure), holomorphic then the vector bundle is a holomorphic vector bundle (this requires the matrix group to...

the research on holomorphic curves in the 20th century and remain central in this subfield. He stated a fundamental theorem on holomorphic curves in complex...

family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector...

curvature theorem for the Quillen metric on the holomorphic determinant of a direct image by a holomorphic proper submersion. This and Bismut—Lebeau's embedding...

{\displaystyle {\mathcal {O}}(D)} can be written as s/f, where s is a holomorphic function on U. Let η be a section over U of ωX. The Poincaré residue...

g = 1 (elliptic curve); and g > 1 (Riemann surfaces with independent holomorphic differentials). In the case of surfaces, the Enriques classification...

} the holomorphic functional calculus allows to define f ( x ) ∈ A {\displaystyle f(x)\in A} for any function f {\displaystyle f} holomorphic in a neighborhood...

led to investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed...

A=e^{i{\frac {\pi }{2}}(I-A)}=e^{-i{\frac {\pi }{2}}(I-A)}} . Adjugate matrix Holomorphic functional calculus Resolvent formalism / Roger A. Horn and Charles R...

connected compact complex manifold of dimension 2 with a nowhere-vanishing holomorphic 2-form. (The latter condition says exactly that the canonical bundle...

} with π + {\displaystyle \pi ^{+}} holomorphic (complex linear) and π − {\displaystyle \pi ^{-}} anti-holomorphic (conjugate linear). For s l ( 2 , C...

George Marinescu the Ferran Sunyer i Balaguer Prize in 2006 for the book "Holomorphic Morse inequalities and Bergman kernels". He was an invited speaker at...

via holomorphic sheaves and their cohomology groups; and for work on foliations. With Chern he worked on Nevanlinna theory, studied holomorphic vector...

by Samuel Goldberg and Shoshichi Kobayashi to allow positivity of the holomorphic bisectional curvature. Inspired by work of René Thom, Frankel and Aldo...

is a complex manifold X. In this case, one considers the algebra of holomorphic functions, i.e., complex-valued differentiable functions. Their ratios...

{\displaystyle \ -{\tfrac {\ \zeta '(s)\ }{\zeta (s)}}\ } differ by a function holomorphic on   ℜ ( s ) = 1   . {\displaystyle \ \Re (s)=1~.} Since, as was shown...

dimensional complex analysis and the topological structure of spaces of Holomorphic functions. He later worked on bounded symmetric domains and spectral...

ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on C ∖ Z ≤ 0 {\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq...

{\displaystyle a} and b {\displaystyle b} . Hellmuth Kneser first proposed a holomorphic construction of the solution of f ( f ( x ) ) = e x {\displaystyle f{\bigl...

MacQuillan gave a new proof of a conjecture of André Bloch (1926) about holomorphic curves in closed subvarieties of Abelian varieties, proved a conjecture...

Mathematicians in Helsinki gave a talk Integral methods and zeros of holomorphic functions. His doctoral students include Jean-Pierre Demailly. "Sous-ensembles...

Sikorav, Jean-Claude (1994), "Some properties of holomorphic curves in almost complex manifolds", Holomorphic curves in symplectic geometry, Progress in Mathematics...

is equal to the eigenvalue of Tn on f. Let f {\displaystyle f} be a holomorphic cusp form with weight ( 2 k + 1 ) / 2 {\displaystyle (2k+1)/2} and character...

rational map f = ΘLΘ−1 from the complex sphere to itself such that Θ is a holomorphic map from a complex torus to the complex sphere and L is an affine map...

zeroth spherical Bessel function. Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in...

equations and 4-manifold topology Yongbin Ruan, Higher genus pseudo-holomorphic curves Edward Witten, Monopoles and four-manifolds John C. Baez, Quantum...

)^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).} As a holomorphic function, sin z is a 2D solution of Laplace's equation: Δ u ( x 1 , x...

series of some smooth function. In complex analysis, however, every holomorphic function is analytic. A function whose Taylor series converges to the...

Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ(f), is either a...

{ a } → C {\displaystyle f:U\smallsetminus \{a\}\to \mathbb {C} } a holomorphic function. The point a {\displaystyle a} is called an essential singularity...

function Holomorphic function Meromorphic function Cauchy–Riemann equations Formal power series Laurent series Theorems Analyticity of holomorphic functions...

functions of a complex variable rather than of a real variable. Indeed, all holomorphic functions are analytic, so that the failure of the function f defined...

an American mathematician, specializing in the functional analysis of holomorphic functions. Forelli received his bachelor's degree from the University...

holomorphic function of τ {\displaystyle \tau } in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function...

simultaneously contains (copies of) a group and its automorphism group Holomorphic functions, the central object of study of complex analysis Biology Teleomorph...

The fact that cusp forms decay at the cusps, rather than merely being holomorphic there as the case with more general modular forms, means that there is...

contributions of the two scientists to the Cauchy and the Dirichlet problem for holomorphic functions of several complex variables, as well as the impact of their...

{\displaystyle \omega } -tameness and (perturbed) J {\displaystyle J} -holomorphic curves: We say that the almost complex structure J {\displaystyle J}...

translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used...

\pper Analytic classification of saddle resonant singular points of holomorphic vector fields on complex plane, J. of Dynamical and Control Systems,...

{\displaystyle \mathbb {C} } to C {\displaystyle \mathbb {C} } (the holomorphic functions) the set of all functions from the natural numbers to the natural...

z_{2})=f_{1}(z_{1})+f_{2}(z_{2})} where f i {\displaystyle f_{i}} are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of...

the value of the function is zero Zero (complex analysis), a zero of a holomorphic function Zero element, generalization of the number zero in algebraic...

Lionel J. (2009), "The Einstein–Weyl equations, scattering maps, and holomorphic disks", Mathematical Research Letters, 16 (2): 291–301, arXiv:0806.3761...

Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space H 2 ( Ω ) {\displaystyle...

invariants, was published in the Journal of Differential Geometry as `A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations'...

, P ( G ) {\displaystyle {\mathcal {P}}(G)} (of continuous, smooth, holomorphic, regular functions) on groups are Hopf algebras in the category (Ste...

the Picard group of Xs.) Sheaf cohomology Chow variety Cartier divisor Holomorphic line bundle Ideal class group Arakelov class group Group-stack Picard...

Differential Equations, Peter J. Olver (2nd ed., 1993, ISBN 978-1-4684-0276-6) Holomorphic Functions and Integral Representations in Several Complex Variables,...

the ring H ( U ) {\displaystyle {\mathcal {H}}(U)} consisting of all holomorphic functions is an integral domain. The same is true for rings of analytic...

two well-behaved functions, in the sense of those two functions being holomorphic. The Karush–Kuhn–Tucker conditions are first-order necessary conditions...

}{2}}\right)\right)^{2g-2}q^{k\beta }} , where β {\displaystyle \beta } is the class of holomorphic curves with genus g, λ {\displaystyle \lambda } is the topological string...

\mathbb {C} :\operatorname {Re} \,z>0\}} such that: f {\displaystyle f} is holomorphic on H {\displaystyle H} ; f ( 1 ) = 1 {\displaystyle f(1)=1} ; f ( z +...

using complex numbers in three dimensions. The basic idea is to use a holomorphic (also called analytic) or meromorphic function f, which maps the physical...

differential equations. He was a pioneer in the study of moduli spaces of holomorphic vector bundles on projective varieties. His work is considered the foundation...

Atiyah studied the problem of the relation between topological and holomorphic vector bundles on projective space. They solved the simplest unknown...

{\displaystyle Y_{-n}(x)=(-1)^{n}Y_{n}(x).} Both Jα(x) and Yα(x) are holomorphic functions of x on the complex plane cut along the negative real axis...

even for complex values of x, because both sides are entire (that is, holomorphic on the whole complex plane) functions of x, and two such functions that...

Clifford J.; Hamilton, Richard S. (1970). "A fixed point theorem for holomorphic mappings". In Chern, Shiing-Shen; Smale, Stephen (eds.). Global analysis...

canonical bundle on C. By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms...

mathematical epidemiologist Laura Ortíz-Bobadilla, Mexican expert on holomorphic foliations Hinke Osinga (born 1969), Dutch expert in dynamical systems...

divisor (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic p-form on X−D. If both ω and dω have a pole of order at most 1 along...

bundle of M {\displaystyle M} is trivial. M {\displaystyle M} has a holomorphic n {\displaystyle n} -form that vanishes nowhere. The structure group...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

f ( z ) w {\displaystyle {\frac {d^{2}w}{dz^{2}}}=f(z)w} where f is holomorphic in a simply-connected region and w is a solution of the differential...

Kähler manifold, and L a holomorphic line bundle on X. Then L is a positive line bundle if and only if there is a holomorphic embedding φ : X → P N {\displaystyle...

the Riemann–Hilbert correspondence. On the other hand, if we work with holomorphic (rather than algebraic) vector bundles with flat connection on a noncompact...

{\displaystyle V} form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V {\displaystyle...

differentiation are ill-conditioned. However, if f {\displaystyle f} is a holomorphic function, real-valued on the real line, which can be evaluated at points...

body of his work involves embedded contact homology, or ECH. ECH is a holomorphic curve model for the Seiberg–Witten Floer homology of a three-manifold...

any holomorphic 1-form on V {\displaystyle V} is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent...

Zakeri : Mating Siegel quadratic polynomials" (PDF). Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical...

Princeton University under Joseph Kohn with thesis Boundary Behavior of Holomorphic Functions on Weakly Pseudoconvex Domains. He is a professor at Purdue...

also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that...

is a holomorphic function, where it is defined. It asks for a meromorphic function f on M such that f − f i {\displaystyle f-f_{i}} is holomorphic on Ui;...

generalization) Mean value theorem Multivariate normal distribution (to do) Holomorphic functions are analytic Pythagorean theorem Quadratic equation Quotient...

Mathematics Magazine, volume 69, no. 2, 1996, p. 126 Serge Colombo: Holomorphic Functions of One Variable. Taylor & Francis 1983, ISBN 0677059507, p...

Milne-Thomson circle theorem and the Milne-Thomson method for finding a holomorphic function are named after him. Milne-Thomson was made a Commander of the...

conclusion that any such curve is a modular curve. A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C)...

doi:10.1007/bf01457933, S2CID 121072397 Sunada, Toshikazu (1978), "Holomorphic equivalence problem for bounded Reinhaldt domains", Mathematische Annalen...

stacks over finite fields. Fixed-point theorems Lefschetz zeta function Holomorphic Lefschetz fixed-point formula Lefschetz, Solomon (1926). "Intersections...

Unit tangent bundle Cotangent bundle Frame bundle Musical isomorphism Holomorphic tangent bundle The disjoint union ensures that for any two points x1...

again assume |x| > |y| and define the powers of x + y and x using a holomorphic branch of log defined on an open disk of radius |x| centered at x. The...

of ‘regular’ for quaternionic functions similar to the definition of holomorphic function by means of an analogue of the Cauchy-Riemann equations. He...

\|x\|^{2}}\qquad \mathrm {Im} \,\tau >0.} The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of...

Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane...

to the non-compact case. This is used to show existence theorems for holomorphic functions. He later worked on pseudogroups and their deformation theory...

bundles; these are the Chern classes. Grothendieck splitting principle for holomorphic vector bundles on the complex projective line H. Blane Lawson and Marie-Louise...

habilitation at Paris-Sud University with a thesis on approximately holomorphic techniques and monodromy invariants in symplectic topology. As a postdoc...

{\displaystyle \mathbb {R} ^{4}} (instantons) correspond bijectively to holomorphic vector bundles on complex projective 3-space C P 3 {\displaystyle \mathbb...

between the partial derivatives called the Cauchy–Riemann equations – see holomorphic functions. Another generalization concerns functions between differentiable...

D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory. Chern–Simons theories are related to many other...

S2CID 119253397. Guilfoyle, B.; Klingenberg, W. (2020). "Fredholm-regularity of holomorphic discs in plane bundles over compact surfaces". Ann. Fac. Sci. Toulouse...

complex structure can be defined by requiring the transition maps to be holomorphic. Smooth frame – Generalization of an ordered basis of a vector spacePages...

boundary circles. This problem can be reduced to that of finding univalent holomorphic maps f, g of the unit disk and its complement into the extended complex...

\|x\|^{2}}\qquad \mathrm {Im} \,\tau >0.} The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of...

can consider polynomials of Grassmann numbers, leading to the idea of holomorphic functions. One can take derivatives of such functions, and then consider...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

JSTOR 2047347 Sankaran, Parameswaran (2003). "A Coincidence Theorem for Holomorphic Maps to G/P". Canadian Mathematical Bulletin. 46 (2): 291–98. arXiv:math/0208062...

completed her doctorate in 1998 from Cornell University; her dissertation, Holomorphic functions and the heat kernel measure on an infinite dimensional complex...

_{X}^{\bullet })=:H_{DR}^{p+q}(X/k)} . Another example comes from the holomorphic log complex on a complex manifold. Let X be a complex algebraic manifold...

Riemann sphere, O ( k ) {\displaystyle {\mathcal {O}}(k)} are the usual holomorphic line bundles over projective space, and the sheaves under consideration...

atlas of charts to the open unit disk such that the transition maps are holomorphic Complex projective space, a projective space with respect to the field...

absolutely for Re ( s ) > 1 {\displaystyle {\text{Re}}(s)>1} and defines a holomorphic function in this domain. For every number field, its ring of integers...

mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers....

complex analysis, a point z {\displaystyle z} on the complex plane where a holomorphic function is undefined, is called a singularity. Some different types...

the Bergman and Szegő projections and their corresponding spaces of holomorphic functions." Bonami is the sister of Georges Nivat, a specialist of Russian...

sequence of holomorphic functions on an open domain. This result has been generalized to normal families of meromorphic functions, holomorphic functions...

integral has to be evaluated. From complex analysis it is known that a holomorphic function can be presented as a Laurent series. From the physics of the...

includes the Virasoro algebra as well as some spin-3 currents. The local holomorphic W primaries are given by 1 , ϵ , σ 1 , σ 2 , ψ 1 , ψ 2 {\displaystyle...

of inner region of }}\Gamma \to \mathbb {C} } is a continuous mapping holomorphic throughout the inner region of Γ {\displaystyle \Gamma } , then ∫ Γ f...

exists on a given closed complex manifold: the first Chern class of the holomorphic tangent bundle must be zero. The necessity of this condition was previously...

in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starlike. Nevanlinna used...

theory (Lagrangian) Affine Gaudin models (Hamiltonian) Six-dimensional holomorphic Chern–Simons theory Exactly solvable statistical lattice models Ising...

useful in analysis, algebra, and geometry when considering germs of holomorphic functions, local rings, and differential forms. J. Arthur also wrote...

some extra structure, we can also take the sheaf of differentiable, or holomorphic functions. Both of these give rise to locally ringed spaces. If X {\displaystyle...

( t ) {\displaystyle Q(t)} is holomorphic in the lower half plane, Q ( t ) − 1 {\displaystyle Q(t)^{-1}} is holomorphic in the lower half plane as well...

{\bar {\partial }}_{A}} consists of the holomorphic sections of E {\displaystyle E} with respect to the holomorphic structure induced by the Dolbeault operator...

inversion formula easily follows. The Segal–Bargmann space is the space of holomorphic functions on Cn that are square-integrable with respect to a Gaussian...

of asterodermids is reminiscent of modern guitarfish. Well-preserved holomorphic ray fossils have been known from Late Jurassic Lagerstätte deposits throughout...

When one considers Bergman space, namely the space of square-integrable holomorphic functions over some domain, then, under general circumstances, one can...

that the complex gamma function and its scalar multiples are the only holomorphic functions on the positive complex half-plane that obey the functional...

numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated...

\mathbb {R} ^{n}} and the space H ( Ω ) {\displaystyle H(\Omega )} of holomorphic functions on an open set Ω ⊂ C n {\displaystyle \Omega \subset \mathbb...

years, these functions were a mystery, but they have become known as the holomorphic parts of harmonic weak Maass forms. Although there are numerous statements...

{\mathbb {R} }\cup \{-\infty \}} is said to be plurisubharmonic if for any holomorphic map φ : Δ → X {\displaystyle \varphi \colon \Delta \to X} the function...

University University of Illinois at Chicago University of Chicago Thesis Holomorphic Families of Rational Maps: Dynamics, Geometry, and Potential Theory (2002)...

real and complex methods: complex analysis gives powerful tools for holomorphic and harmonic functions, while boundary convergence and estimates in Lp...

3-form. This construction was extended to equivariant K-theory and to holomorphic K-theory by Mathai and Stevenson. Bundle gerbes have also appeared in...

about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial. Let X {\displaystyle X} and Y {\displaystyle...

his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme...

forth. Often, a complex domain serves as the domain of definition for a holomorphic function. In the study of several complex variables, the definition of...

complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations. It is thus...

Goulden, I. P. (1997). "Transitive factorisations into transpositions and holomorphic mappings on the sphere" (PDF). Proceedings of the American Mathematical...

with a Kähler metric) such that weakly pseudoconvex, and let F be a holomorphic line bundle over X equipped with a singular hermitian metric of weight...

constant maps are harmonic morphisms. Holomorphic functions in the complex plane are harmonic morphisms. Holomorphic functions in the complex vector space...

ISBN 978-3-662-57264-1. McDuff, Dusa; Salamon, Dietmar (2012) [2004]. J-holomorphic Curves and Symplectic Topology. Colloquium Publications. Vol. 52 (2nd ed...

Kähler manifolds to families of Hodge structures. Let f : X → B be a holomorphic submersive morphism. For a point b of B, we denote the fiber of f over...

mathematics—the dynamics of Kleinian group actions and the iteration of holomorphic and anti-holomorphic maps in complex dynamics. His results have reshaped our understanding...

is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product...

{\displaystyle \tau _{2}} ; M 1 {\displaystyle {\mathcal {M}}_{1}} , holomorphic to the moduli space of the torus, is any fundamental domain for the modular...

algebra A ( D ) {\displaystyle A(\mathbf {D} )} consists of functions holomorphic in the open unit disk D ⊆ C {\displaystyle D\subseteq \mathbb {C} } and...

the state space leads to irrational functions (which are however still holomorphic). In discrete-time the transfer function is given in terms of the state-space...

injective metric space containing the points of a given metric space. The holomorphically convex hull is a generalization of similar concepts to complex analytic...

MR 1362832 Cohen, Ralph L.; Lupercio, Ernesto; Segal, Graeme B. (1999). "Holomorphic spheres in loop groups and Bott periodicity". Asian Journal of Mathematics...

functions. The Académie des Sciences awarded him the Poncelet Prize in 1882. Holomorphic function Timeline of abelian varieties Félix, Lucienne (2008), "Briot...

group acts on the space of square integrable functions. In the theta, or holomorphic, model, the Heisenberg group acts on a Hilbert space of entire functions...

the theory is the number of pseudo holomorphic maps f : M → X in the sense of Gromov (they are ordinary holomorphic maps if X is a Kähler manifold). If...