Monogenic function

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

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 $$ f(x) $$ which is monogenic $$ \forall \zeta \in B $$, is said to be monogenic on $$ B $$, and if $$ B $$ is a domain of $$ \mathbb{C}$$, 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 $$ \mathbb{C} $$, can show a weakened form of analyticity)