Complex Analysis/Complex Functions/Complex Derivatives
With the concept of the limit of a complex function now established, we can now introduce the differentiability of a complex function.
Definition: Let f be a complex valued function defined in a neighborhood of . Then the derivative of f at is given by:
, where is a complex number.
Provided that this limit exists, f is said to be differentiable at .
Here, is any complex number, so it may approach zero in a number of different directions. However, for this limit to exist, it must reach a unique limit f'() independent of how approaches zero. For this reason, the complex conjugation is nowhere differentiable. To see this consider the limit
For this limit is and for this limit is , so there does not exist a unique complex limit.