Monogenic function explained

A monogenic^{[1] [2]} 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'(\zeta) exists and is finite, with:$$f\text{'}(\backslash zeta)\; =\; \backslash lim\_\backslash frac$$

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 \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)

Monogenic term was coined by Cauchy.^[3]

Notes and References

1. Web site: Monogenic function . Encyclopedia of Math . 15 January 2021.
2. Web site: Monogenic Function. Wolfram MathWorld . 15 January 2021.
3. Book: A history of analysis . 2003 . American Mathematical Society ; London Mathematical Society . 978-0-8218-2623-2 . Jahnke . H. N. . History of mathematics . Providence, RI : [London] . 229.