Holomorphic function

This article describes holomorphy as a property of functions. For another meaning, see Holomorphy of a group.

Holomorphy (from Gr. ὅλος holos, "whole" and μορφή morphe, "form") is a property of certain complex-valued functions treated in function theory (a branch of mathematics). A function $f\colon U\to {\mathbb {C}}$ with an open set $U\subseteq \mathbb {C}$ is called holomorphic if it is complex differentiable at every point of In particular, in older literature such functions are also called regular.

Even though the definition is analogous to real differentiability, it turns out in function theory that holomorphy is a very strong property. Namely, it produces a variety of phenomena that have no counterpart in the real. For example, every holomorphic function is arbitrarily often (continuously) differentiable and can be locally developed into a power series at any point.

A rectangular grid is transformed into its image using the holomorphic function

Definitions

Let $U\subseteq \mathbb {C}$ an open subset of the complex plane and $z_{0}\in U$ a point of this subset. A function $f\colon U\to {\mathbb {C}}$ is called complex differentiable at the point $z_{0}$ if the limit

$\lim _{h\to 0}{\frac {f(z_{0}+h)-f(z_{0})}{h}}$

exists. It is then called $f'(z_{0})$ .

The function is called holomorphic at the point $z_{0}$ if there $z_{0}$ exists a neighborhood of in which is complex differentiable. If is holomorphic on all of , then called holomorphic. Further, if $U={\mathbb {C}}$ , then is called an entire function.

Notes

Relationship between complex and real differentiability

$\mathbb {C}$ is naturally a two-dimensional real vector space with canonical basis $\{1,i\}$ and so a function $f\colon U\to {\mathbb {C}}$ on an open set $U\subseteq \mathbb {C}$ also be examined for its total differentiability in the sense of multidimensional real analysis. As is well known, (total) differentiable in $z_{0}$ , if an $\mathbb {R}$ -linear mapping exists such that.

$f(z_{0}+h)=f(z_{0})+L(h)+r(h)$

where is a function with

$\lim _{h\to 0}{\frac {r(h)}{|h|}}=0$

is. Now we see that the function is complex differentiable in $z_{0}$ exactly if it is totally differentiable there and even $\mathbb {C}$ is -linear. The latter is a strong condition. It means that the representation matrix of with respect to the canonical basis $\{1,i\}$ has the form

${\begin{pmatrix}a&-b\\b&a\end{pmatrix}}$

has.

Cauchy-Riemann differential equations

→ Main article: Cauchy-Riemann partial differential equations

If we now decompose a function $f\left(x+iy\right)=u\left(x,y\right)+i\,v\left(x,y\right)$ into its real and imaginary parts with real functions u,v , then the total derivative has as representation matrix the Jacobi matrix

${\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.$

Consequently, the function is complex differentiable if and only if it is real differentiable and for u,v the Cauchy-Riemann differential equations

${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}$

$\displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$

are fulfilled.