Holomorphic function/Criteria

Introduction

Holomorphy of a function $f\colon U\to \mathbb {C}$ at a point $z_{0}\in U$ is a neighborhood property of $z_{0}$ . There are numerous criteria in complex analysis that can be used to verify holomorphy. Let $U\subseteq \mathbb {C}$ be a domain as a subset of the complex plane and $z_{0}\in U$ a point in this subset.

Animation - Visualization of the Mapping

The animation shows the function $f(z)={\frac {1}{z}}$ . In the animation, $z$ is shown in blue, and the corresponding image point $f(z)$ is shown in red. The point $z$ and $f(z)$ are represented in $\mathbb {C} {\widetilde {=}}\mathbb {R} ^{2}$ . The $y$ -axis represents the imaginary part of the complex numbers $z$ and $f(z)$ . The blue point $z$ moves along the path $\gamma (t):=t\cdot (\cos(t)+i\cdot \sin(t))$

Animation

Complex Differentiability

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 with $h\in \mathbb {C}$ . This is denoted as $f'(z_{0})$ .

Holomorphy

A function $f\colon U\to \mathbb {C}$ is called holomorphic at the point $z_{0}$ if there exists a neighborhood $U_{0}\subseteq U$ of $z_{0}$ such that $f$ is complex differentiable in $U_{0}$ . If $f$ is holomorphic on all of $U$ , it is simply called holomorphic. If additionally $U=\mathbb {C}$ , $f$ is called an entire function.

Holomorphy Criteria

Let $f:U\to \mathbb {C}$ be a function where $U\subseteq \mathbb {C}$ is a domain, then the following properties of the complex-valued function $f$ are equivalent:

(HK1) Once Complex Differentiable

The function $f$ is once complex differentiable on $U$ .

(HK2) Arbitrarily Often Complex Differentiable

The function $f$ is arbitrarily often complex differentiable on $U$ .

(HK3) Cauchy-Riemann Differential Equations

The real and imaginary parts satisfy the Cauchy-Riemann equations and are at least once continuously real-differentiable on $U$ .

(HK4) Locally Expansible in Power Series

The function can be locally expanded in a complex power series on $U$ .

(HK5) Path Integrals 0

The function $f$ is continuous, and the path integral of the function over any closed contractible path vanishes (i.e., the winding number of the path integral for all points outside of $U$ is 0).

(HK6) Cauchy Integral Formula

The function values inside a circular disk can be determined from the function values on the boundary using the Cauchy integral formula.

(HK7) Cauchy-Riemann Operator

$f$ is real differentiable, and ${\frac {\partial f}{\partial {\bar {z}}}}=0$ , where ${\frac {\partial }{\partial {\bar {z}}}}$ is the Cauchy-Riemann operator defined by ${\frac {\partial }{\partial {\bar {z}}}}:={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)$ .

Exercises

Let $a,z_{0}\in \mathbb {C}$ be chosen arbitrarily, and assume that $a\neq z_{0}$ . Now, develop the function $f(z):={\frac {1}{z-a}}$ for $z\in \mathbb {C} \setminus {a}$ in a power series around $z_{0}\in \mathbb {C}$ and show that the following holds: $f(z)={\frac {1}{z-a}}=\sum _{n=0}^{\infty }-{\frac {1}{(a-z_{0})^{n+1}}}\cdot (z-z_{0})^{n}$

Calculate the radius of convergence of the power series! Explain why the radius of convergence depends on $a,z_{0}\in \mathbb {C}$ in this way and cannot be larger!

It is not true in real analysis that the existence of a once differentiable function implies that the function is infinitely differentiable. Consider the function $g(x):=x\cdot |x|$ defined on all of $\mathbb {R}$ .

Explain how the central theorem of Complex Analysis from criterion 1 leads to criterion 2!

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Date: 12/10/2024 9:56pm

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