Complex Analysis/Complex Functions/Complex Derivatives

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Complex differentiability[edit | edit source]

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.


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


We choose

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


a contradiction.

The Cauchy–Riemann equations[edit | edit source]

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

are well-defined, differentiable at and satisfy the equations

These equations are called the Cauchy-Riemann equations.


1. We prove well-definedness of .

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

Then we have:

From these equations follows the existence of , since for example

exists due to lemma 2.2.3.

The proof for

and the existence of we leave for exercise 2.

Holomorphic functions[edit | edit source]

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 | edit source]

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