holomorphic function

1. Complex Differentiable Function

Let $E$ be a subset in $\mathbb{C}$ and $f$ a complex-valued function on $E$. The function $f$ is complex differentiable at the point $z_0$, where $z_0$ is a limit point of $E$, iff

$$\underset{z\to z_0,z\in E\setminus \{z_0\}}{\lim }\frac{f(z)-f(z_0)}{z-z_0}=L\in \mathbb{C}$$

The limit is called the derivative of $f$ at $z_0$ and is denoted by $f^{\prime }(z_0)$.

Other equivalent definitions:
(1)

$$\underset{\begin{array}{c}h\to 0\\ h\ne 0\end{array}}{\lim }\frac{f(z_0+h)-f(z_0)}{h}=L\in \mathbb{C}$$

(2)

$$\underset{z\to z_0,z\in E\setminus \{z_0\}}{\lim }\frac{|f(z)-(f(z_0)+L(z-z_0))|}{|z-z_0|}=0$$

1.1. Rules of Arithmetic of complex differentiable functions

If $f$ and $g$ are complex differentiable in $\Omega$, then:

• $f+g$ is complex differentiable in $\Omega$ and $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$.
• $fg$ is complex differentiable in $\Omega$ and $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$.
• If $g(z_0)\ne 0$, then $f/g$ is complex differentiable at $z_0$ and $(f/g{)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}.$

Moreover, if $f:\Omega \to U$ and $g:U\to \mathbb{C}$ are complex differentiable, the chain rule holds
$(g\circ f{)}^{\prime }(z)=g^{\prime }(f(z))f^{\prime }(z)\text{for all }z\in \Omega .$

For the quotient rule, set $A(z)=f(z)-f(z_0)-f^{\prime }(z_0)(z-z_0)$ and $B(z)=g(z)-g(z_0)-g^{\prime }(z_0)(z-z_0)$.
Given that: $\frac{A(z)}{z-z_0}\to 0,\frac{B(z)}{z-z_0}\to 0$ as $z\to z_0$ (by definition)
We can write the following ratio:

$$\frac{f(z)}{g(z)}-\left(\frac{f(z_0)}{g(z_0)}+L(z-z_0)\right)=\frac{f(z)g(z_0)-f(z_0)g(z)-L(z-z_0)g(z)g(z_0)}{g(z)g(z_0)}$$

$f(z)=f(z_0)+f^{\prime }(z_0)(z-z_0)+A(z)$ and $g(z)=g(z_0)+g^{\prime }(z_0)(z-z_0)+B(z)$.
Substituting into the ratio, we get:

$$\frac{(f^{\prime }(z_0)g(z_0)-f(z_0)g^{\prime }(z_0)-Lg(z)g(z_0))(z-z_0)+A(z)g(z_0)-f(z_0)B(z)}{g(z)g(z_0)}$$

$L=\frac{f^{\prime }(z_0)g(z_0)-f(z_0)g^{\prime }(z_0)}{g(z_0{)}^2}$.
We have $f^{\prime }(z_0)g(z_0)-f(z_0)g^{\prime }(z_0)=Lg(z_0{)}^2$.
Therefore
the ratio is:

$$\begin{array}{r}\frac{(Lg(z_0{)}^2-Lg(z)g(z_0))(z-z_0)+A(z)g(z_0)-f(z_0)B(z)}{g(z)g(z_0)}\\ =\frac{(Lg(z_0)(g(z_0)-g(z)))(z-z_0)+A(z)g(z_0)-f(z_0)B(z)}{g(z)g(z_0)}\end{array}$$

It is easy to see that the numerator tends to 0 as $z\to z_0$ and the denominator tends to $g(z_0{)}^2$.

2. Holomorphic Function

A function is holomorphic on an open set $U$ if it is complex differentiable at every point of $U$. A function $f$ is holomorphic at a point $z_0$ if it is holomorphic on some neighbourhood of $z_0$. A function is holomorphic on some non-open set $A$ if it is holomorphic at every point of $A$.

if $f$ is holomorphic in all of $\mathbb{C}$ we say that $f$ is entire.

3. Properties

3.1. Cauchy-Riemann equations

Let $z,z_0\in \Omega$, $z=x+iy$, $z_0=x_0+i{y}_0$,
if $f(x+iy)=u(x,y)+iv(x,y)$ be holomorphic in $\Omega$. Then

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

proof:

$$\begin{array}{rl}{f}^{\prime }(z_0)& =\underset{\begin{array}{c}y\to y_0\\ y\ne y_0\end{array}}{\lim }\frac{f(x_0+iy)-f(x_0+i{y}_0)}{i(y-y_0)}\\ & =\underset{\begin{array}{c}y\to y_0\\ y\ne y_0\end{array}}{\lim }\frac{u(x_0,y)+iv(x_0,y)-u(x_0,y_0)-iv(x_0,y_0)}{i(y-y_0)}\\ & =\underset{\begin{array}{c}y\to y_0\\ y\ne y_0\end{array}}{\lim }\frac{u(x_0,y)-u(x_0,y_0)}{i(y-y_0)}+i\underset{\begin{array}{c}y\to y_0\\ y\ne y_0\end{array}}{\lim }\frac{v(x_0,y)-v(x_0,y_0)}{i(y-y_0)}\\ & =\frac{1}{i}\frac{\partial u}{\partial y}+\frac{\partial v}{\partial y}\\ & =-i\frac{\partial u}{\partial y}+\frac{\partial v}{\partial y}\end{array}$$

similarly,

$$\begin{array}{rl}{f}^{\prime }(z_0)& =\underset{\begin{array}{c}x\to x_0\\ x\ne x_0\end{array}}{\lim }\frac{f(x+i{y}_0)-f(x_0+i{y}_0)}{x-x_0}\\ & =\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}\end{array}$$

Because the two limits above are both equal to $f^{\prime }(z_0)$, we have

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

3.2. Derivative and Jacobian Properties of Holomorphic Functions

$$\frac{\partial }{\partial z}=\frac{1}{2}\left(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right)\text{and}\frac{\partial }{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right).$$

If $f$ is holomorphic at $z_0$, then

$$\frac{\partial f}{\partial \bar{z}}(z_0)=0\text{and}{f}^{\prime }(z_0)=\frac{\partial f}{\partial z}(z_0)=2\frac{\partial u}{\partial z}(z_0).$$

Also, if we write $F(x,y)=f(z)$, then $F$ is differentiable in the sense of real variables, and

$$\det J_F(x_0,y_0)=|f^{\prime }(z_0){|}^2.$$