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A monogenic 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. 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 in a manner such that functions which are monogenic over non-connected subsets of ${\displaystyle \mathbb {C} }$, can show a weakened form of analyticity)

Monogenic term was coined by Cauchy.

References

1. ^ "Monogenic function". Encyclopedia of Math. Retrieved 15 January 2021.
2. ^ "Monogenic Function". Wolfram MathWorld. Retrieved 15 January 2021.
3. Jahnke, H. N., ed. (2003). A history of analysis. History of mathematics. Providence, RI : : American Mathematical Society ; London Mathematical Society. p. 229. ISBN 978-0-8218-2623-2.

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