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 ...

with a single finite

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

. 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

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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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

\mathbb

, 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

\mathbb

, can show a weakened form of analyticity)

References

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