As compared with real differentiable functions which are defined on real intervals, complex functions are defined for subsets of the complex plane. In this section, we will define differentiability for complex functions and also study their properties. Many of these properties will be similar to those of real differentiable functions.
However, as the complex plane allows two degrees of freedom (as compared to only one degree of freedom for the real axis), we will be able to significantly extend the concept of differentiability, with astonishing consequences.