Monogenic function

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

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

The term monogenic was coined by Cauchy.