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A monogenic function is a complex function with a single finite derivative. More precisely, a function f {\displaystyle f} defined on A ⊆ C {\displaystyle A\subseteq \mathbb {C} } is called monogenic at ζ ∈ A {\displaystyle \zeta \in A} , if f ′ {\displaystyle f'} exists and is finite, with:
Alternatively, it can be defined as the above limit having the same value for all paths. Functions can either have a single derivative or infinitely many derivatives , with no intermediate cases. Furthermore, a function f {\displaystyle f} 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