Monogenic function

A monogenic^[1][2] function is a complex function with a single finite derivative. More precisely, a function ${\displaystyle f(z)}$ defined on ${\displaystyle A\subseteq \mathbb {C} }$ is called monogenic at ${\displaystyle \zeta \in A}$, if ${\displaystyle f'(\zeta )}$ exists and is finite, with: $${\displaystyle f'(\zeta )=\lim _{z\to \zeta }{\frac {f(z)-f(\zeta )}{z-\zeta }}}$$

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.^[2] Furthermore, a function ${\displaystyle f(x)}$ which is monogenic ${\displaystyle \forall \zeta \in B}$, is said to be monogenic on ${\displaystyle B}$, and if ${\displaystyle B}$ is a domain of ${\displaystyle \mathbb {C} }$, then it is analytic as well (The notion of domains can also be generalized ^[1] in a manner such that functions which are monogenic over non-connected subsets of ${\displaystyle \mathbb {C} }$, can show a weakened form of analyticity)

The term monogenic was coined by Cauchy.^[3]

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