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

concerned with analytic functions of a complex variable, that is, holomorphic functions. The concept can be extended to functions of several complex variables...

analysis, a complex-valued function f {\displaystyle f} of a complex variable z {\displaystyle z} : is said to be holomorphic at a point a {\displaystyle...

heading. As in complex analysis of functions of one variable, which is the case n = 1, the functions studied are holomorphic or complex analytic so that, locally...

antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of...

complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated...

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

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

on this class of functions. In several ways, the harmonic functions are real analogues to holomorphic functions. All harmonic functions are analytic, that...

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

criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function f defined on an open set D in...

analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily characterized. For the case of an analytic function with...

In mathematics, a modular form is a holomorphic function on the complex upper half-plane, H {\displaystyle {\mathcal {H}}} , that roughly satisfies a functional...

L2(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H2(∂D) uniquely extend to holomorphic functions in D, and the...

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...

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 f is...

in 1844), states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a...

complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. The function f : z ↦ 2 z + z 2 {\displaystyle...

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

derivative of a holomorphic function; every nowhere-vanishing holomorphic function f {\displaystyle f} on G {\displaystyle G} has a holomorphic logarithm;...

removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such...

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 neighborhood...

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 by Frigyes...

\gamma ^{*}(s,z),} extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in z and s. It follows from the...

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 injectively...

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 subsets...

= {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient...

the modular lambda function, usually denoted by λ {\textstyle \lambda } , and which performs, using modern terminology, the holomorphic universal covering...

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. Suppose...

function is a bijective holomorphic function whose inverse is also holomorphic. Formally, a biholomorphic function is a function ϕ {\displaystyle \phi }...