We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.
MSC 2010: 33E12, 30A10, 30D15, 30E15We consider some families of 3-index generalizations of the classical Mittag-Le²er functions and study the behaviour of these functions in domains of the complex plane. First, some inequalities in the complex plane and on its compact subsets are obtained. We also prove an asymptotic formula for the case of "large" values of the indices of these functions. Similar results have also been obtained by the author for the classical Bessel functions and their Wright's...
In this work we consider the Dunkl operator on the complex plane, defined by We define a convolution product associated with denoted and we study the integro-differential-difference equations of the type , where is a sequence of complex numbers and is a measure over the real line. We show that many of these equations provide representations for particular classes of entire functions of exponential type.
We give a pure complex variable proof of a theorem by Ismail and Stanton and apply this result in the field of integer-valued entire functions. Our proof rests on a very general interpolation result for entire functions.