Complex Analysis/Complex Functions/Complex Derivatives

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.


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

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

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

  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   .