# Complex Analysis/Complex Functions/Complex Derivatives

## Complex differentiability

Let us now define what complex differentiability is.

Definition 2.3.1:

Let ${\displaystyle S\subseteq \mathbb {C} }$, let ${\displaystyle f:S\to \mathbb {C} }$ be a function and let ${\displaystyle z_{0}\in S}$. ${\displaystyle f}$ is called complex differentiable at ${\displaystyle z_{0}}$ if and only if there exists a ${\displaystyle w\in \mathbb {C} }$ such that:

${\displaystyle \lim _{z\to z_{0} \atop z\in S}{\frac {f(z)-f(z_{0})}{z-z_{0}}}=w}$

Example 2.3.2

The function

${\displaystyle f:\mathbb {C} \to \mathbb {C} ,f(z)={\overline {z}}}$

is nowhere complex differentiable.

Proof:

Let ${\displaystyle z_{0}\in \mathbb {C} }$ be arbitrary. Assume that ${\displaystyle f}$ is complex differentiable at ${\displaystyle z_{0}}$, i. e. that

${\displaystyle \lim _{z\to z_{0} \atop z\in \mathbb {C} }{\frac {{\overline {z}}-{\overline {z_{0}}}}{z-z_{0}}}}$

exists.

We choose

${\displaystyle A:=\{z\in \mathbb {C} |\Re z=\Re z_{0}\}}$

and

${\displaystyle B:=\{z\in \mathbb {C} |\Im z=\Im z_{0}\}}$

. Due to lemma 2.2.3, which is applicable since of course ${\displaystyle \mathbb {C} }$ is open, we have:

${\displaystyle \lim _{z\to z_{0} \atop z\in A}{\frac {{\overline {z}}-{\overline {z_{0}}}}{z-z_{0}}}=\lim _{z\to z_{0} \atop z\in \mathbb {C} }{\frac {{\overline {z}}-{\overline {z_{0}}}}{z-z_{0}}}=\lim _{z\to z_{0} \atop z\in B}{\frac {{\overline {z}}-{\overline {z_{0}}}}{z-z_{0}}}}$

But

${\displaystyle \lim _{z\to z_{0} \atop z\in A}{\frac {{\overline {z}}-{\overline {z_{0}}}}{z-z_{0}}}=\lim _{z\to z_{0} \atop z\in A}{\frac {\Re (z-z_{0})-i\Im (z-z_{0})}{\Re (z-z_{0})+i\Im (z-z_{0})}}=-1}$

and

${\displaystyle \lim _{z\to z_{0} \atop z\in B}{\frac {{\overline {z}}-{\overline {z_{0}}}}{z-z_{0}}}=\lim _{z\to z_{0} \atop z\in B}{\frac {\Re (z-z_{0})-i\Im (z-z_{0})}{\Re (z-z_{0})+i\Im (z-z_{0})}}=1}$

${\displaystyle ////}$

## The Cauchy-Riemann equations

We can define a natural bijective function from ${\displaystyle \mathbb {C} }$ to ${\displaystyle \mathbb {R} ^{2}}$ as follows:

${\displaystyle \Phi (x+iy):=(x,y)}$

In fact, ${\displaystyle \Phi }$ is a vector space isomorphism between ${\displaystyle \mathbb {C} ^{1}}$ and ${\displaystyle \mathbb {R} ^{2}}$.

The inverse of ${\displaystyle \Phi }$ is given by

${\displaystyle \Phi ^{-1}:\mathbb {R} ^{2}\to \mathbb {C} ,\Phi ^{-1}(x,y)=x+iy}$

Theorem and definitions 2.3.3:

Let ${\displaystyle O\subseteq \mathbb {C} }$ be open, let ${\displaystyle f:O\to \mathbb {C} }$ be a function and let ${\displaystyle z_{0}=x_{0}+iy_{0}\in O}$. If ${\displaystyle f}$ is complex differentiable at ${\displaystyle z_{0}}$, then the functions

${\displaystyle u:\Phi (O)\to \mathbb {R} ,u(x,y)=\Re f(x+iy)}$

and

${\displaystyle v:\Phi (O)\to \mathbb {R} ,v(x,y)=\Im f(x+iy)}$

are well-defined, differentiable at ${\displaystyle (x_{0},y_{0})}$ and satisfy the equations

${\displaystyle \partial _{x}u(x_{0},y_{0})=\partial _{y}v(x_{0},y_{0})}$

and

${\displaystyle \partial _{y}u(x_{0},y_{0})=-\partial _{x}v(x_{0},y_{0})}$

. These equations are called the Cauchy-Riemann equations.

Proof:

1. We prove well-definedness of ${\displaystyle u}$ and ${\displaystyle v}$.

Let ${\displaystyle (x,y)\in \Phi (O)}$. We apply the inverse function on both sides to obtain:

${\displaystyle x+iy\in \Phi ^{-1}(\Phi (O))=O}$

, where the last equality holds since ${\displaystyle \Phi }$ is bijective (for any bijective ${\displaystyle f:S_{1}\to S_{2}}$ we have ${\displaystyle f^{-1}(f(S_{3}))=f(f^{-1}(S_{3}))=S_{3}}$ if ${\displaystyle S_{3}\subseteq S_{1}}$; see exercise 1).

3. We prove differentiability of ${\displaystyle u}$ and ${\displaystyle v}$ and the Cauchy-Riemann equations.

We define

${\displaystyle S_{1}:=\{z\in \mathbb {C} |\Re z=\Re z_{0}\}\cap O}$

and

${\displaystyle S_{2}:=\{z\in \mathbb {C} |\Im z=\Im z_{0}\}\cap O}$

Then we have:

{\displaystyle {\begin{aligned}\partial _{x}u(x_{0},y_{0})&=\lim _{x\to x_{0}}{\frac {u(x,y_{0})-u(x_{0},y_{0})}{x-x_{0}}}&\\&=\lim _{x\to x_{0}}{\frac {\Re f(x+iy_{0})-\Re f(x_{0}+iy_{0})}{x-x_{0}}}&\\&=\Re \left(\lim _{x\to x_{0}}{\frac {f(x+iy_{0})-f(x_{0}+iy_{0})}{x-x_{0}}}\right)&{\text{continuity of }}\Re \\&=\Re \left(\lim _{z\to z_{0} \atop z\in S_{2}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}\right)&\\&=\Re \left(\lim _{z\to z_{0} \atop z\in S_{1}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}\right)&{\text{lemma 2.2.3}}\\&=\Re \left(\lim _{y\to y_{0}}{\frac {f(x_{0}+iy)-f(x_{0}+iy_{0})}{iy-iy_{0}}}\right)&\\&=\Re \left((-i)\lim _{y\to y_{0}}{\frac {f(x_{0}+iy)-f(x_{0}+iy_{0})}{y-y_{0}}}\right)&i^{-1}=-i\\&=\Im \left(\lim _{y\to y_{0}}{\frac {f(x_{0}+iy)-f(x_{0}+iy_{0})}{y-y_{0}}}\right)&\\&=\partial _{y}v(x_{0},y_{0})\end{aligned}}}

From these equations follows the existence of ${\displaystyle \partial _{x}u(x_{0},y_{0})}$ and ${\displaystyle \partial _{y}v(x_{0},y_{0})}$, since for example

${\displaystyle \lim _{z\to z_{0} \atop z\in S_{2}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}}$

exists due to lemma 2.2.3.

The proof for

${\displaystyle \partial _{y}u(x_{0},y_{0})=-\partial _{x}v(x_{0},y_{0})}$

and the existence of ${\displaystyle \partial _{y}u(x_{0},y_{0})}$ and ${\displaystyle \partial _{x}v(x_{0},y_{0})}$ we leave for exercise 2.

${\displaystyle ////}$

## Holomorphic functions

Definitions 2.3.4:

Let ${\displaystyle S\subseteq \mathbb {C} }$ and let ${\displaystyle f:S\to \mathbb {C} }$ be a function. We call ${\displaystyle f}$ holomorphic if and only if for all ${\displaystyle z_{0}\in S}$, ${\displaystyle f}$ is differentiable at ${\displaystyle z_{0}}$. In this case, the function

${\displaystyle f':S\to \mathbb {C} ,f'(z_{0})=\lim _{z\to z_{0} \atop z\in S}{\frac {f(z)-f(z_{0})}{z-z_{0}}}}$

is called the complex derivative of ${\displaystyle f}$. We write ${\displaystyle H(S)}$ for the set of holomorphic functions defined on ${\displaystyle S}$.

## Exercises

1. Let ${\displaystyle S_{1},S_{2},S_{3}}$ be sets such that ${\displaystyle S_{3}\subseteq S_{1}}$, and let ${\displaystyle f:S_{1}\to S_{2}}$ be a bijective function. Prove that ${\displaystyle f^{-1}(f(S_{3}))=f(f^{-1}(S_{3}))=S_{3}}$.
2. Let ${\displaystyle O\subseteq \mathbb {C} }$ be open, let ${\displaystyle f:O\to \mathbb {C} }$ be a function and let ${\displaystyle z_{0}=x_{0}+iy_{0}\in O}$. Prove that if ${\displaystyle f}$ is complex differentiable at ${\displaystyle z_{0}}$, then ${\displaystyle \partial _{y}u(x_{0},y_{0})}$ and ${\displaystyle \partial _{x}v(x_{0},y_{0})}$ exist and satisfy the equation ${\displaystyle \partial _{y}u(x_{0},y_{0})=-\partial _{x}v(x_{0},y_{0})}$.

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