Complex Analysis/Complex Functions/Complex Derivatives
Let us now define what complex differentiability is.
Let , let be a function and let . is called complex differentiable at if and only if there exists a such that:
is nowhere complex differentiable.
Let be arbitrary. Assume that is complex differentiable at , i. e. that
. Due to lemma 2.2.3, which is applicable since of course is open, we have:
, a contradiction.
The Cauchy-Riemann equations
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 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.
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.
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 .
- 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 .