Complex Analysis/Complex Functions/Complex Derivatives

Complex differentiabilityEdit

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

 

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

 

But

 

a contradiction. 

The Cauchy–Riemann equationsEdit

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.

Proof

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 functionsEdit

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   .

ExercisesEdit

  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   .

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