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^{n}} are holomorphically separable, in particular, all domains in C n {\displaystyle \mathbb {C} ^{n}} and all Stein manifolds. A holomorphically separable...

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

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

studies holomorphic functions, which are differentiable functions of a complex variable. In contrast with a differentiable real function, a holomorphic function...

polynomially convex hull contains the holomorphically convex hull. The domain G {\displaystyle G} is called holomorphically convex if for every compact subset...

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...

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

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

under the name holomorphically symplectic manifolds. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic...

{\displaystyle U\smallsetminus \{a\}} ⁠. We say f {\displaystyle f} is holomorphically extendable over U {\displaystyle U} if such a g {\displaystyle g} exists...

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

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

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

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...

consequence, any complete Kähler manifold of constant holomorphic sectional curvature is holomorphically isometric to a non-singular quotient of one of these...

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...

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...

v_{j})\right|\leq \|\sum _{i}v_{i}\|^{2},} which follows because the sum depends holomorphically on S and T, which are unitary on the boundary. Let S denote the unit...

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...

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...

{\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...

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

(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})}...

of the boundary of its disc of convergence and still fail to extend holomorphically beyond that disc. For example, if α > 0 {\displaystyle \alpha >0} is...

abelian varieties turn out to be exactly those complex tori that can be holomorphically embedded into a complex projective space. Abelian varieties defined...

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...

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

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

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 X} is holomorphically convex, i.e. for every compact subset K ⊂ X {\displaystyle K\subset X} , the so-called holomorphic hull (or envelope...

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

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...

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

manifolds. A manifold X {\displaystyle X} is Stein if it is holomorphically convex and holomorphically separable (see the article on Stein manifolds for the...

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

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

the outer boundary such that f ( z ) {\displaystyle f(z)} cannot be holomorphically extended to those points; giving rise to a Riemann-Hilbert problem...

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

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

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...

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

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

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

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...

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...

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

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...

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...

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

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

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

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...

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

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...

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...

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...

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

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

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

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

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

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...

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...

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

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

S2CID 122535410, Zbl 0038.23502 Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces". American Journal of Mathematics. 82 (4):...

following. Every holomorphic vector bundle E {\displaystyle {\mathcal {E}}} on C P 1 {\displaystyle \mathbb {CP} ^{1}} is holomorphically isomorphic to a...

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

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

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...

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

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

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...

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...

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...

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

{\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...

fundamental cycles, Fukaya categories of certain manifolds and counting holomorphic curves, and for his contributions to other areas of geometry and topology...

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

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

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

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

{\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...

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

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...

_{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...

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

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

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...

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

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

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

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

this means that there is a genuine bound on the size of discs mapping holomorphically into X. In these terms, Schwarz's lemma says that the unit disc D is...

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

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...

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

1051/0004-6361/202554998. Bonneau, Pierre; Mazzilli, Emmanuel (2023). "Almost holomorphic curves in real analytic hypersurfaces". arXiv:2311.01298 [math.CV]. Patwardhan...

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

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

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....

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...

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

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

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...

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

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

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

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

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

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

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...

143–160. Dineen, Seán; Patyi, Imre; Venkova, Milena "Inverses depending holomorphically on a parameter in a Banach space". J. Funct. Anal. 237 (2006), no....

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...

{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...

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...

first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic. The dimension of the space of differentials of the first kind, by means...

include: Removable singularities - in which the function can be extended holomorphically to z {\displaystyle z} Poles - in which the function can be extended...

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

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

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...

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

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

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...

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...

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...

{\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...

{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...

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

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

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

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...

that each Xb is Kähler, but that there is a Kähler class that varies holomorphically in b. In other words, assume there is a class ω in H2(X, Z) such that...

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...

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

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

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

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...

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

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

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...

complex structures. If one further imposes that the canonical bundle is holomorphically trivial, meaning that it is global sections which are closed forms...

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

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...

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...

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...

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

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

approximated by Artin L-functions. Using the fact that Artin L-functions are holomorphic in a neighbourhood of the line ℜ ( s ) = 1 {\displaystyle \Re (s)=1}...

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

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

{\displaystyle a} to the nearest point where the function cannot be continued holomorphically. In many common examples this is the distance to the nearest singularity...

\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...

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...

described by mathematician Anthony Philips as leading to a "revival of holomorphic dynamics after 60 years of stagnation." 1971 Oswald Veblen Prize in Geometry...

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

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

Carathéodory's theorem (convex hull) Choquet theory Helly's theorem Holomorphically convex hull Integrally-convex set John ellipsoid Pseudoconvexity Radon's...

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

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...

{\displaystyle \mathbb {C} } is not holomorphic at 0 {\displaystyle 0} , since for complex functions, holomorphicity at a point implies analyticity at that...

suppose U is an open set in C and f : U → C is a function that has a holomorphic antiderivative F on U. Then for every curve γ : [a, b] → U, the curve...

of holomorphy. Ω {\displaystyle \Omega } is holomorphically convex. There exists a function holomorphic on Ω {\displaystyle \Omega } that cannot be analytically...

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...

(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...

{\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...

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

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

quotient space V/L, where V is the dual of the vector space of all global holomorphic differentials on C and L is the lattice of all elements of V of the form...

{\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...

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

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

set is a domain of holomorphy (holomorphically convex), a convex tube is also a domain of holomorphy. So the holomorphic envelope of any tube is equal...

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...

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

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

function of a complex variable is holomorphic if its derivative exists as a complex limit at each point of its domain. Holomorphic functions are much more rigid...

many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function...

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...

real variable to ones of a complex variable gave rise to the concept of holomorphic functions and the inception of complex analysis as another distinct subdiscipline...

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...

\mathbb {C} ,} and f : U → C {\displaystyle f:U\to \mathbb {C} } is a holomorphic function, then f {\displaystyle f} has an antiderivative F {\displaystyle...

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

the harmonic series. It can be extended by analytic continuation to a holomorphic function on all complex numbers except x = 1 {\displaystyle x=1} , where...

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...

=-J(t\partial /\partial t).} It is nowhere vanishing. It commutes with all holomorphic Killing vectors on the cone and in particular with all isometries of...

3-dimensional Euclidean space is isometric to a Bryant surface by a holomorphic parameterization analogous to the (Euclidean) Weierstrass–Enneper parameterization...

doi:10.4171/114-1/14. ISBN 978-3-03719-114-9. Bogomolov, F. A. (1979). "Holomorphic Tensors and Vector Bundles on Projective Varieties". Izvestiya Akademii...

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...

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

mathematician Jacques Feldbau, and she documented the genesis of the modern holomorphic dynamics, with detailed portraits of the main protagonists: Pierre Fatou...

\operatorname {Re} (s)>1} ⁠, but the zeta function can be extended to a holomorphic function defined on C ∖ { 1 } {\displaystyle \mathbb {C} \setminus \{1\}}...

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

Aganagić, Mina; Vafa, Cumrun (2000), Mirror symmetry, D-branes and counting holomorphic discs, arXiv:hep-th/0012041, Bibcode:2000hep.th...12041A Aganagić, Mina;...

torsion). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An almost Hermitian structure is given by an almost...

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

m . {\displaystyle n>m.} Let g : X → Y {\displaystyle g:X\to Y} be a holomorphic map and let y ∈ im ( g ) {\displaystyle y\in {\text{im}}(g)} be such...

therefore be analytically continued (or continuously extended, or holomorphically extended) to all of V by Riemann's analytic continuation theorem. So...

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...

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

Misha (1999). "Topological invariants of dynamical systems and spaces of holomorphic maps I". Mathematical Physics, Analysis and Geometry. 2 (4): 323–415...

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

\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...

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...

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

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...

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

w(z_{0})=0} . Let κ R {\displaystyle \kappa R} be the Bergman kernel for holomorphic (1, 0) forms on R. We define B R ( z ) | d w | 2 := κ R ( z ) | V z 0...

} 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...

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

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

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

best known for his work on the theory of integral representations for holomorphic functions of several variables, notably for discovering the Bochner–Martinelli...

1, ∞, there are usually two special solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicial equation...

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...

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...

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...

)^{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...

{2}}e^{i(3\pi /4)}=-1+i=-{\sqrt {-2i}}.} The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly...

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

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

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...

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

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

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

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...

and imaginary parts u {\displaystyle u} and v {\displaystyle v} of a holomorphic function. This first order system of partial differential equations has...

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

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...

invertible sheaf is (the sheaf of sections of) a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically...

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

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...

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

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

\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...

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

39.1.0207. S2CID 118792515. Fritzsche, K.; Grauert, H. (2002). From Holomorphic Functions to Complex Manifolds. Springer. p. 34. ISBN 9780387953953....

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

scalar manifold. The second is the superpotential, which is an arbitrary holomorphic function W ( ϕ ) {\displaystyle W(\phi )} that fixes a number of aspects...

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

theorems limit the number of possible lacunary values of certain types of holomorphic functions. Clark, Douglas N., ed. (1999). Dictionary of Analysis, Calculus...

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...

Daniel; Rassias, Michael Th. (eds.). Advancements in Complex Analysis – Holomorphic Approximation. Springer Nature. pp. 133–192. arXiv:1802.03924. doi:10...

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

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

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...

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...

Other automorphic forms associated to these congruence subgroups are the holomorphic modular forms, which can be interpreted as cohomology classes on the...

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

the supervision of David Gross. His Ph.D. thesis on Four Dimensional Holomorphic Theories was defended in 1996. There he introduced a four dimensional...

Scientific career Fields Mathematics Institutions University of Notre Dame Thesis Holomorphic R-Torsion for Lie Groups  (1973) Doctoral advisor Isadore Singer...

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...

{\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} are known holomorphic functions, the functions Ξ + ( α ) {\displaystyle \Xi _{+}(\alpha )}...

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...

manifolds, complex algebraic varieties, and complex analytic varieties, and holomorphic vector bundles and coherent sheaves over these spaces. Special examples...

family of this genus, Dictyosporiaceae, was created to accommodate a holomorphic group of Dothideomycetes, including 12 genera that are saprobes on decaying...

{\displaystyle z\mapsto \int (T_{z}f)g\,d\mu _{2}} is continuous on S and holomorphic on S for all simple functions  f  and g. For some constant k < π, the...

case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below. A complete linear system on X {\displaystyle...

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...

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

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

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...

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

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...

theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact...

morphisms), the category of compact Riemann surfaces (with non-constant holomorphic maps as morphisms), and the opposite of the category of algebraic function...

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...

of the importance of the sheaf of regular functions or the sheaf of holomorphic functions. Grothendieck elegantly defined and characterized sheaf cohomology...

function are both holomorphic in the critical strip, where the imaginary part of t is between −1/2 and 1/2, that the Z function is holomorphic in the critical...

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

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

metric. This associates a pseudo-metric to any complex manifold, in a holomorphically invariant way. This sets up the important notion of Kobayashi hyperbolicity...

{\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...

following. Let K be the canonical bundle on M. The dimension of H0(M,Km), holomorphic sections of Km, is denoted by Pm(M), called m-genus. Let N ( M ) = {...

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

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...

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...

function is a meromorphic function on the whole complex plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1. In 1737...

the Laplace transform gives a one-to-one correspondence between the holomorphic functions which, for some ⁠ σ ∈ R {\displaystyle \sigma \in \mathbb {R}...

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

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 ...

transcendental methods such as, for example, topology and the theory of holomorphic functions of one complex variable are required to prove the theorem....

additive group of A, with the given parametrization. Again these act holomorphically on A2 and the action passes to the quotient X. On A the action is given...

{\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...

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

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...

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...

just θ ¯ ± {\displaystyle {\overline {\theta }}^{\pm }} ) but depend holomorphically on the twisted superpotential, and vice versa for the B model. A number...

Giovanni; Gerretsen, Johan (1960), Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen, MR 0113988...

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

to prove this one can for instance find invariants coming from pseudo-holomorphic curves. Perhaps the simplest partial differential relation is for the...

) {\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...

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

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...

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...

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

a generalization of the theorem is stated such: Let f : Ω → C be a holomorphic function on the open convex set Ω, and let a and b be distinct points...

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...

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

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...

points have finite D'Angelo type has analytic properties, with respect to holomorphic functions, that are quite similar to strictly pseudoconvex domains. David...

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

formulated the Bieberbach conjecture, stating a necessary condition for a holomorphic function to map the open unit disc injectively into the complex plane...

integral domains, it is the product of quotient fields. For the ring of holomorphic functions on an open set D of complex numbers, the total quotient ring...

because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not. The following...

second Betti number is 2, the second Chern number vanishes, and the holomorphic Euler characteristic vanishes. They were classified by (Bombieri & Mumford...

sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces. Given a dynamical...

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...

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

Journal of Mathematics. with Christophe Soulé: Direct images of Hermitian holomorphic bundles Bull. Amer. Math. Soc. vol. 15, 1986, 209–212 doi:10...

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

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...

and Arnaud Beauville that M [ n ] {\displaystyle M^{[n]}} is also holomorphically symplectic. This is not very difficult to see, e.g., for n = 2 {\displaystyle...

angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface—in which case the surface is called a Riemann...

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

of the eigenvalues. A similar technique works more generally with the holomorphic functional calculus, using A − 1 = Q Λ − 1 Q − 1 {\displaystyle \mathbf...

Misha (1999). "Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I". Mathematical Physics, Analysis and Geometry. 2 (4): 323–415...

variety V of complex dimension n it is the number of linearly independent holomorphic n-forms to be found on V. This definition, as the dimension of H0(V,Ωn)...

of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language...

main theorem.) A regular map between complex algebraic varieties is a holomorphic map. (There is actually a slight technical difference: a regular map...

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...

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

theory: for instance, in Knapp (2002), G is a complex Lie group and V is a holomorphic representation of G with an open dense orbit. The two approaches are...

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

do postdoctoral research. He works with one-dimensional dynamics and holomorphic functions. Since 2003 he has worked as a researcher for the Centre National...

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

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

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

complex fields satisfying a property known as holomorphy, which allows holomorphic quantities to be exactly computed. This makes supersymmetric models useful...

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

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...

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...

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

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

Analysis, and Applications, p. 196 and Klaus Fritzsche, Hans Grauert, From Holomorphic Functions to Complex Manifolds, p. 168), in medicine (Effect of immersion...

that is ω | L = 0 {\displaystyle \left.\omega \right|_{L}=0} . The holomorphic volume form Ω ∈ Ω n , 0 ( X ) {\displaystyle \Omega \in \Omega ^{n,0}(X)}...

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...

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...

'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...

In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common...

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

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...

addition, he equips these Riemann surfaces with Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X {\displaystyle X} ...

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

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

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...

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

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...

commutative algebra. See also Jet (mathematics). In the development of holomorphic functions of a complex variable z = x + i y, sometimes a complex function...

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...

Matematicheskii Sbornik. 194 (10): 3–26. Akbarov, Sergei S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic...

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

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...

related topics (correspondences on curves, cyclic coverings, bundles of holomorphic forms). ... De Franchis introduced and used implicitly some of the most...

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...

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

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...

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...

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...

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

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

singularities at various points (i.e. points at which it is not a regular holomorphic function, such as branch points, essential singularities or poles). A...

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

f} is a fundamental one. Removable singularity – Undefined point on a holomorphic function which can be made regular Mathematical singularity – Point where...

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

function of several complex variables that is holomorphic in each variable separately is itself holomorphic. It was introduced by William Fogg Osgood in...

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

mapping α ↦ x + α {\displaystyle \alpha \mapsto x_{+}^{\alpha }} is a holomorphic function from the right half-plane to the topological vector space of...

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

embedded case (see Intersection theory for details). Let z0 be a root of a holomorphic function f, and let n be the least positive integer such that the nth...

splitting up Hk(V(C), C) into subspaces Hp,q according to the number p of holomorphic differentials dzi wedged to make up α (the cotangent space being spanned...

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

\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...

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

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

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...

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

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

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

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...

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

Xiaonan Ma, the Ferran Sunyer i Balaguer Prize in 2006 for their book "Holomorphic Morse inequalities and Bergman kernels". Marinescu was awarded the Romanian...

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

Z . {\displaystyle \log(z)\ =\ w\ +\ 2\pi i\mathbf {Z} .} Given any holomorphic function on an open subset of the complex plane C, its analytic continuation...

character on X. Ideally the moduli spaces should be a critical sets of holomorphic Chern–Simons functions and the Donaldson–Thomas invariants should be...

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

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

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

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...

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...

g^{-1}(x)Ag(x)-g^{-1}(x){\frac {dg(x)}{dx}}} are regarded as being equivalent for any holomorphic gauge transformation g(x). (It is thus most natural to regard a Fuchsian...

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...

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...

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...

The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler differentials: κ = ⋀ n   Ω X . {\displaystyle \kappa ={\textstyle...

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

(or equivalently, compact), then the degree of L is determined by the holomorphic Euler characteristics of X and S: deg(L) = χ(X,OX) − 2χ(S,OS). The canonical...

Misha (1999). "Topological invariants of dynamical systems and spaces of holomorphic maps: I". Mathematical Physics, Analysis and Geometry. 2 (4): 323–415...

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...

i p ) = e g ( x + i p ) {\displaystyle F(x+ip)=e^{g(x+ip)}} for some holomorphic function g {\displaystyle g} . But in order for F {\displaystyle F} to...

than the upper half-plane by discrete groups. Siegel modular forms are holomorphic functions on the set of symmetric n × n {\displaystyle n\times n} matrices...

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

But it can be shown that the resolvent function R {\displaystyle R} is holomorphic on its domain. By the vector-valued version of Liouville's theorem, this...

Vietnam team leader for several International Mathematical Olympiads. Holomorphic mappings on Banach analytic manifolds, in Func. Analyz i ego Priloz....

exceptional set as the Zariski closure of the union of images of non-trivial holomorphic maps from C to X. Brody's definition of a hyperbolic variety is that...

convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. Sums and products of analytic functions...

( 2 , C ) {\displaystyle \operatorname {SL} (2,\mathbb {C} )} ), and holomorphic maps. Similarly, using an alternate metric completion of ⁠ Q {\displaystyle...

an almost-complex manifold. The zero set of this section consists of holomorphic maps. If the d-bar operator can be shown to be transverse to the zero-section...

1090/s0002-9904-1961-10661-7. MR 0131663. Bers, Lipman; Ehrenpreis, Leon (1964). "Holomorphic convexity of Teichmüller spaces". Bull. Amer. Math. Soc. 70 (6): 761–764...

with reassessment of Netsepoye hawesi and comments on the morphology of holomorphic petalodonts". Paleontological Journal. 48 (9): 1003–1014. Bibcode:2014PalJ...

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

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...

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

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

algebraic condition under which an almost complex structure arises from a holomorphic coordinate atlas.[NN57] The Newlander-Nirenberg theorem is now considered...

Fatou set of f are both completely invariant under iterations of the holomorphic function f: f − 1 ( J ⁡ ( f ) ) = f ( J ⁡ ( f ) ) = J ⁡ ( f ) , {\displaystyle...

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

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

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

the fertile leaf is different from the sterile leaves. In dimorphic (holomorphic) ferns, the two types of leaves are morphologically distinct. The fertile...

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...

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

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...

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...

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...

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...

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...

formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups. When...

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

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

(1847). A holomorphic elliptic function is constant. This is the original form of Liouville's theorem and can be derived from it. A holomorphic elliptic...

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

open domain is pseudoconvex. Analytic polyhedron Eugenio Elia Levi Holomorphically convex hull Stein manifold Bremermann, H. J. (1956). "Complex Convexity"...

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

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

{\displaystyle \operatorname {sinc} } ⁠ is an even entire function (holomorphic over the entire complex plane), ⁠ Si {\displaystyle \operatorname {Si}...

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...

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

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...

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

\Gamma } , the function ν ( γ , z ) {\displaystyle \nu (\gamma ,z)} is a holomorphic function of z ∈ H {\displaystyle z\in \mathbb {H} } . For all z ∈ H {\displaystyle...

constructed, including one constructed combinatorially by Chekanov and using holomorphic discs by Eliashberg. This Chekanov-Eliashberg invariant yields an invariant...

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

with complex geometry, which includes Kähler manifolds and (stable) holomorphic vector bundles. Simon Donaldson and Peter Kronheimer both obtained their...

the drive-in theater's projection booth, Doc Green gives Mr. Bunty a Holomorphic Projector to replace the movie projector as Kade plans to find a copy...

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

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

irreducible noncompact Kähler manifold admits isometries which are not holomorphic only if the holonomy group is the symplectic group Sp ( n ) {\displaystyle...

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

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...

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...

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

{\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...

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...

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...

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

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

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

are often consequences of the fact that certain objects must have a holomorphic dependence on the quantum fields and coupling constants. In this case...

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...

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

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

Appell–Humbert theorem of Paul Émile Appell and Georges Humbert, classifies the holomorphic line bundles on an abelian surface by cocycle data. 1894 Die Entwicklung...

the ring O ( C n ) {\displaystyle {\mathcal {O}}(\mathbf {C} ^{n})} of holomorphic functions on Cn (functions of n complex variables.) Given some differential...

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

identified the sheaf cohomology groups, for the sheaf associated with the holomorphic tangent bundle, that carried the basic data about the dimension of the...

as the monodromy groups of a Belyi surface – a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity (a Belyi...

algebra H o l ( C ) {\displaystyle \mathrm {Hol} (\mathbb {C} )} of holomorphic functions on C {\displaystyle \mathbb {C} } . Convolution algebra of...

{O}}(U)} of all complex-valued holomorphic functions on U. The uniqueness of analytic continuation says that given two holomorphic functions on a connected...

0|W_{t}\not \in D\}.} If f : D → C {\textstyle f:D\to \mathbb {C} } is a holomorphic function which is not constant, such that f ( 0 ) = 0 {\textstyle f(0)=0}...

{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...

modularity theorem. Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve...

\|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...

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

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...

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

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

^{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...

mathematical physics, four-dimensional Chern–Simons theory, also known as semi-holomorphic or semi-topological Chern–Simons theory, is a quantum field theory initially...

MR 0009635. Mandelbrojt, Szolem; MacLane, Gerald R. (1947). "On functions holomorphic in a strip region, and an extension of Watson's problem". Trans. Amer...

{\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...

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...

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...

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...

 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...

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

converges to ƒ in the Lp norm. The original proof uses properties of holomorphic functions and Hardy spaces, and another proof, due to Salomon Bochner...

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

(CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions. While a CFT might conceivably exist only on a given Riemann...

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

system of orthogonal polynomials for the space of square-integrable holomorphic functions over some domain—that is, a Bergman space. In this case, the...

({\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...

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

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

many) of holomorphic functions, and so an oriented coordinate atlas consisting of isothermal coordinate charts may be viewed as a holomorphic coordinate...

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