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...

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...

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

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

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...

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

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...

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

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...

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

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...

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...

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

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

= {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...

\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...

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

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...

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

exponential function, namely the restriction to the image L ⁡ ( U ) {\displaystyle \operatorname {L} (U)} . Since the exponential function is holomorphic (that...

proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely...

{\displaystyle z_{0}\in \mathbb {C} } . However, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it...

versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces...

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

as may be seen by the ratio test. The function defined by this series can be extended to a holomorphic function defined on all complex numbers with a...

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