Premium
Wiman‐Valiron theory for higher dimensional polynomial Cauchy‐Riemann equations
Author(s) -
De Almeida R.,
Kraußhar R.S.
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4591
Subject(s) - mathematics , cauchy–riemann equations , cauchy distribution , polynomial , cauchy problem , iterated function , context (archaeology) , pure mathematics , initial value problem , operator (biology) , cauchy's integral formula , projection (relational algebra) , mathematical analysis , paleontology , biochemistry , chemistry , repressor , algorithm , gene , transcription factor , biology
In this paper, we introduce different kinds of growth orders for the set of entire solutions to the most general framework of higher‐dimensional polynomial Cauchy‐Riemann equations∏ i = 1 p( D − λ i )k if = 0 , where D : = ∂ f ∂ x 0+ ∑ i = 1 ne i∂ f ∂ x iis the hypercomplex Cauchy‐Riemann operator, λ i are arbitrarily chosen nonzero complex constants, and k i are arbitrarily chosen positive integers. The core ingredient is a projection formula that establishes a relation to the k i ‐monogenic component functions, which are null‐solutions to iterates of the Cauchy‐Riemann operator that we studied in earlier works. Furthermore, we briefly outline the analogies of the Lindelöf‐Pringsheim theorem in this context.