Monogeneity set

The set of all derived numbers (Dini derivatives, cf. Dini derivative) of a given function of a complex variable at a given point. More precisely, let $ E $ be a set in the complex plane $ \mathbf C $, let $ \zeta $ be a non-isolated point of $ E $ and let $ f ( z) $ be a complex function of $ z \in E $. A complex number $ a $( proper or equal to $ \infty $) is called a derived number (or Dini derivative) of $ f $ at $ \zeta $ relative to $ E $ if there is a sequence $ z _ {n} \in E $ with the properties: $ z _ {n} \neq \zeta $, $ z _ {n} \rightarrow \zeta $,

$$ \frac{f ( z _ {n} ) - f ( \zeta ) }{z _ {n} - \zeta } \rightarrow a \ \ \textrm{ as } n \rightarrow \infty . $$

The set $ \mathfrak M ( \zeta , f , E ) $ of all derived numbers of $ f $ at $ \zeta $ relative to $ E $ is called the monogeneity set of $ f $ at $ \zeta $ relative to $ E $( see [1]). The set $ \mathfrak M ( \zeta , f , E ) $ consists of a unique finite point $ a $ if and only if $ f ( z) $ is a monogenic function at $ \zeta $ relative to $ E $ and $ f _ {E} ^ { \prime } ( \zeta ) = a $. The set $ \mathfrak M ( \zeta , f , E ) $ is always closed, and for each closed set $ A $ in the extended complex plane $ \overline{\mathbf C}\; $, each set $ E \subset \mathbf C $ and each finite non-isolated point $ \zeta $ of $ E $, there is a function $ f ( z) $, $ z \in E $, such that $ \mathfrak M ( \zeta , f , E ) = A $. If $ \zeta $ is an interior point of $ E $, then for any function $ f ( z) $ that is continuous in a neighbourhood of $ \zeta $, the set $ \mathfrak M ( \zeta , f , E ) $ is closed and connected (a continuum) in $ \overline{\mathbf C}\; $ and, conversely, for any continuum $ K \subset \overline{\mathbf C}\; $ there is a function $ f ( z) $, continuous in a neighbourhood of $ \zeta $, for which $ \mathfrak M ( \zeta , f , E ) = K $. If $ f ( z) = f ( x + i y ) = u ( x , y ) + i v ( x , y ) $ is differentiable with respect to the set of real variables $ ( x , y ) $ at an interior point $ \zeta = \xi + i \eta $ of $ E $, then $ \mathfrak M ( \zeta , f , E ) $ is the circle $ \gamma ( r , c ) = \{ {w } : {| w - c | = r } \} $( possibly degenerating into a point, $ r = 0 $) with centre $ c = \partial f ( \zeta ) / \partial z $ and radius $ r = | \partial f ( \zeta ) / \partial \overline{z}\; | $, where

$$ \frac{\partial f }{\partial z } = \frac{1}{2} \left ( \frac{\partial f }{\partial x } - i \frac{\partial f }{\partial y } \right ) = \frac{1}{2} \left ( \frac{\partial u }{\partial x } + \frac{\partial v }{\partial y } \right ) + \frac{i}{2} \left ( \frac{\partial v }{\partial x } - \frac{\partial u }{\partial y } \right ) , $$

$$ \frac{\partial f }{\partial \overline{z}\; } = \frac{1}{2} \left ( \frac{\partial f }{\partial x } + i \frac{\partial f }{\partial y } \right ) = \frac{1}{2} \left ( \frac{\partial u }{\partial x } - \frac{\partial v }{\partial y } \right ) + \frac{i}{2} \left ( \frac{\partial v }{\partial x } + \frac{\partial u }{\partial y } \right ) $$

are the so-called formal derivatives. The converse is also true: Each circle is the monogeneity set for some function $ f $, differentiable with respect to $ ( x , y ) $, at a given interior point $ \zeta $ of $ E $.

If $ f ( z) $ is continuous in a domain $ G $, then at almost every $ \zeta \in G $ the set $ \mathfrak M ( \zeta , f , G ) $ is either a circle $ \gamma ( r , c ) $, $ 0 \leq r < \infty $, or is $ \overline{\mathbf C}\; $( see [2]). In the general case of an arbitrary (not necessarily measurable) set $ E $ and an arbitrary (not necessarily measurable) finite function $ f ( z) $, $ z \in E $, at almost every point $ \zeta \in E $ one of the following three cases holds: a) $ \mathfrak M ( \zeta , f , E ) = \gamma ( r , c ) $, $ c \in \mathbf C $, $ 0 \leq r < \infty $; b) $ \mathfrak M ( \zeta , f , E ) = \overline{\mathbf C}\; $; or c) $ \mathfrak M ( \zeta , f , E ) = \gamma ( r , c ) \cup \infty $, $ c \in \mathbf C $, $ 0 \leq r < \infty $. Here, a) holds at almost every differentiability point of $ f ( z) = f ( x + i y ) $ with respect to $ ( x , y ) \in E $ and one of the first two cases holds at almost every continuity point of $ f ( z) $. Each of the cases a)–c) may be realized individually at almost every point $ \zeta \in E $.

For some natural generalizations to the multi-dimensional case see [4].

References

[1]	V.S. Fedorov, "The work of N.N. Luzin on the theory of functions of a complex variable" Uspekhi Mat. Nauk , 7 : 2 (1952) pp. 7–16 (In Russian)
[2]	Yu.Yu. Trokhimchuk, "Continuous mappings and monogeneity conditions" , Moscow (1963) (In Russian)
[3]	E.P. Dolzhenko, "On the derived numbers of complex functions" Izv. Akad. Nauk SSSR Ser. Mat. , 26 (1962) pp. 347–360 (In Russian)
[4]	A.V. Bondar, "Continuous operator conformal mappings" Ukr. Math. J. , 32 : 3 (1980) pp. 207–212 Ukrain. Mat. Zh. , 32 : 3 (1980) pp. 314–322