# Complex Analysis/Complex Functions/Complex Derivatives

## Contents

## Complex differentiability[edit]

Let us now define what complex differentiability is.

**Definition 2.3.1**:

Let , let be a function and let . is called **complex differentiable at ** if and only if there exists a such that:

**Example 2.3.2**

The function

is nowhere complex differentiable.

**Proof**:

Let be arbitrary. Assume that is complex differentiable at , i. e. that

exists.

We choose

and

. Due to lemma 2.2.3, which is applicable since of course is open, we have:

But

and

, a contradiction.

## The Cauchy-Riemann equations[edit]

We can define a natural bijective function from to as follows:

In fact, is a vector space isomorphism between and .

The inverse of is given by

**Theorem and definitions 2.3.3**:

Let be open, let be a function and let . If is complex differentiable at , then the functions

and

are well-defined, differentiable at and satisfy the equations

and

. These equations are called the **Cauchy-Riemann equations**.

**Proof**:

1. We prove well-definedness of and .

Let . We apply the inverse function on both sides to obtain:

, where the last equality holds since is bijective (for any bijective we have if ; see exercise 1).

3. We prove differentiability of and and the Cauchy-Riemann equations.

We define

and

Then we have:

From these equations follows the existence of and , since for example

exists due to lemma 2.2.3.

The proof for

and the existence of and we leave for exercise 2.

## Holomorphic functions[edit]

**Definitions 2.3.4**:

Let and let be a function. We call **holomorphic** if and only if for all , is differentiable at . In this case, the function

is called the **complex derivative of **. We write for the set of holomorphic functions defined on .

## Exercises[edit]

- Let be sets such that , and let be a bijective function. Prove that .
- Let be open, let be a function and let . Prove that if is complex differentiable at , then and exist and satisfy the equation .