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.

Feb 14, 2026 · 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 …

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

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 …

However, the function g is holomorphic in ̄Dρ (its singular point ζ = z lies outside this set) and hence the Cauchy theorem for multiply connected domains may be applied.

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.

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).

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.

A function that is analytic on a region A is called holomorphic on A. A function that is analytic on A except for a set of poles of finite order is called meromorphic on A.