A monogenic function is a

complex function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic g ...

with a single finite

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

. More precisely, a function

f(z)

defined on

A \subseteq \mathbb

is called monogenic at

\zeta \in A

, if

f'(\zeta)

exists and is finite, with:

f'(\zeta) = \lim_\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 A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...

\mathbb

, then it is

analytic Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemical ...

as well (The notion of domains can also be generalized in a manner such that functions which are monogenic over non-connected subsets of

\mathbb

, can show a weakened form of analyticity) The term monogenic was coined by

Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

References

{{mathanalysis-stub Mathematical analysis Functions and mappings