A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a …

A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

Usually, we speak of functions as holomorphic on (open) sets, rather than at points, for when we consider the behavior of a function at a point, we prefer to consider it in the context of the points nearby.

Dec 3, 2025 · A synonym for analytic function, regular function, differentiable function, complex differentiable function, and holomorphic map (Krantz 1999, p. 16). The word derives from the Greek …

Functions that are C-differentiable at all points of an open set D ⊂ C are clearly also holomorphic at all points z ∈ D. We say that such functions are holomorphic in D and denote their collection by O(D).

In complex analysis, a holomorphic function is a complex differentiable function. The condition of complex differentiability is very strong, and leads to an especially elegant theory of calculus for these …

F3(z + h) F3(z) lim does not exist; h!0 h and so F3(z) = z is not a holomorphic function.

Holomorphic functions (also called analytic functions) usually refer to functions that are infinitely differentiable; They are a big part of complex analysis (the study of functions of complex numbers).

May 28, 2025 · A holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point.

Finally, we note the following lemma, which allows us to extend bounds on a holomorphic function that hold at the edges of a vertical strip to the interior of the strip.