Monogenic function

A monogenic{{cite web |title=Monogenic function |url=http://encyclopediaofmath.org/index.php?title=Monogenic_function&oldid=47887 |website=Encyclopedia of Math |access-date=15 January 2021}}{{cite web |title=Monogenic Function|url=https://mathworld.wolfram.com/MonogenicFunction.html |website=Wolfram MathWorld |access-date=15 January 2021}} function is a complex function with a single finite derivative.

More precisely, a function $f(z)$ defined on $A\; \backslash subseteq\; \backslash mathbb\{C\}$ is called monogenic at $\backslash zeta\; \backslash in\; A$, if $f\text{'}(\backslash zeta)$ exists and is finite, with:

$$f\text{'}(\backslash zeta)\; =\; \backslash lim\_\{z\backslash to\backslash zeta\}\backslash frac\{f(z)\; -\; f(\backslash zeta)\}\{z\; -\; \backslash zeta\}$$

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

The term monogenic was coined by Cauchy.{{Cite book |title=A history of analysis |date=2003 |publisher=American Mathematical Society ; London Mathematical Society |isbn=978-0-8218-2623-2 |editor-last=Jahnke |editor-first=H. N. |series=History of mathematics |location=Providence, RI : [London] |page=229}}