Holomorphic function/Criteria

Introduction

Holomorphy of a function $f:U\to ℂ$ 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 $U\subseteq ℂ$ 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 $ℂ\widetilde{=}{ℝ}^2$. The $y$-axis represents the imaginary part of the complex numbers $z$ and $f(z)$. The blue point $z$ moves along the path $\gamma (t):=t\cdot (\cos (t)+i\cdot \sin (t))$

Error creating thumbnail:

Complex Differentiability

A function $f:U\to ℂ$ is called complex differentiable at the point $z_0$ if the limit $\underset{h\to 0}{\lim }\frac{f(z_0+h)-f(z_0)}{h}$ exists with $h\in ℂ$. This is denoted as $f^{\prime }(z_0)$.

Holomorphy

A function $f:U\to ℂ$ 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 $U=ℂ$, $f$ is called an entire function.

Holomorphy Criteria

Let $f:U\to ℂ$ be a function where $U\subseteq ℂ$ 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 $\frac{\partial f}{\partial \overline{z}}=0$, where $\frac{\partial }{\partial \overline{z}}$ is the Cauchy-Riemann operator defined by $\frac{\partial }{\partial \overline{z}}:=\frac{1}{2}(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y})$.

Exercises

Let $a,z_0\in ℂ$ be chosen arbitrarily, and assume that $a\ne z_0$. Now, develop the function $f(z):=\frac{1}{z-a}$ for $z\in ℂ\setminus a$ in a power series around $z_0\in ℂ$ and show that the following holds: $$f(z)=\frac{1}{z-a}=\sum _{n=0}^{\mathrm{\infty }}-\frac{1}{(a-z_0{)}^{n+1}}\cdot (z-z_0{)}^n$$

Calculate the radius of convergence of the power series! Explain why the radius of convergence depends on $a,z_0\in ℂ$ 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 $ℝ$.

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

See also

• Complex Analysis
• Complex Analysis/Quiz

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• Date: 12/10/2024 9:56pm