A monogenic function is a complex function with a single finite derivative. More precisely, a function f ( z ) {\displaystyle f(z)} defined on A ⊆ C {\displaystyle A\subseteq \mathbb {C} } is called monogenic at ζ ∈ A {\displaystyle \zeta \in A} , if f ′ ( ζ ) {\displaystyle f'(\zeta )} exists and is finite, with:
f ′ ( ζ ) = lim z → ζ f ( z ) − f ( ζ ) z − ζ {\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. Furthermore, a function f ( x ) {\displaystyle f(x)} which is monogenic ∀ ζ ∈ B {\displaystyle \forall \zeta \in B} , is said to be monogenic on B {\displaystyle B} , and if B {\displaystyle B} is a domain of C {\displaystyle \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 C {\displaystyle \mathbb {C} } , can show a weakened form of analyticity) The term monogenic was coined by Cauchy.
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