Lesson plan
Monogenic function
A monogenic function is a complex function with a single finite derivative. More precisely, a function f(z) defined on A \subseteq \mathbb{C} is called monogenic at \zeta \in A , if f'(\zeta) exists and is finite, with: 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 f(x) which is monogenic \forall \zeta \in B , is said to be monogenic on B , and if B is a domain of \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 \mathbb{C} , can show a weakened form of analyticity) The term monogenic was coined by Cauchy.
Article summary
A monogenic function is a complex function with a single finite derivative. More precisely, a function f(z) defined on A \subseteq \mathbb{C} is called monogenic at \zeta \in A , if f'(\zeta) exists and is finite, with: 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 f(x) which is monogenic \forall \zeta \in B , is said to be monogenic on B , and if B is a domain of \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 \mathbb{C} , can show a weakened form of analyticity) The term monogenic was coined by Cauchy.
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Discussion questions
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