Monogenic function

A monogenic function is a complex function with a single finite derivative. More precisely, a function [math]\displaystyle{ f(z) }[/math] defined on [math]\displaystyle{ A \subseteq \mathbb{C} }[/math] is called monogenic at [math]\displaystyle{ \zeta \in A }[/math], if [math]\displaystyle{ f'(\zeta) }[/math] exists and is finite, with: [math]\displaystyle{ f'(\zeta) = \lim_{z\to\zeta}\frac{f(z) - f(\zeta)}{z - \zeta} }[/math]

Alternatively, it can be defined as the above limit having the same value for all paths. Functions can either have a single derivative (monogenic) or infinitely many derivatives (polygenic), with no intermediate cases. Furthermore, a function [math]\displaystyle{ f(x) }[/math] which is monogenic [math]\displaystyle{ \forall \zeta \in B }[/math], is said to be monogenic on [math]\displaystyle{ B }[/math], and if [math]\displaystyle{ B }[/math] is a domain of [math]\displaystyle{ \mathbb{C} }[/math], then it is analytic as well (The notion of domains can also be generalized in a manner such that functions which are monogenic over non-connected subsets of [math]\displaystyle{ \mathbb{C} }[/math], can show a weakened form of analyticity)