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

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...

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...

Zariski's theory of formal holomorphic functions. Algebraic geometry based on formal schemes is called formal algebraic geometry. Formal schemes are usually...

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

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

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

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...

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

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

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...

incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case...

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...

= {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...

approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch...

Schwarz's lemma, Lindelöf principle, analogues and generalizations". A holomorphic function on an open subset of the complex plane is called univalent if it...

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...

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...

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...

f} is a holomorphic function, then φ ( z ) = log ⁡ | f ( z ) | {\displaystyle \varphi (z)=\log \left|f(z)\right|} is a subharmonic function if we define...

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...

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

requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences...

arguments. The functions sinh z and cosh z are then holomorphic. Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:...

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

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

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...

semi-continuous function f : X → R ∪ { − ∞ } {\displaystyle f\colon X\to {\mathbb {R} }\cup \{-\infty \}} is said to be plurisubharmonic if for any holomorphic map...

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

1}(s-1)\zeta (s)=1.} Thus the Riemann zeta function is a meromorphic function on the whole complex plane, which is holomorphic everywhere except for a simple pole...