Charakterisierung des Cotangens mit Replikativität
type: Item
instance of: scholarly article
title: Charakterisierung des Cotangens mit Replikativität
zbMATH Open document ID: 0589.39008
DOI: 10.1007/BF01172154
author: Rolf Jäger
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155164
review text: The function \(x\to \cot (\pi x)\) \((0<x<1)\) is a solution of the system \[ f(x)=(1/k)\sum^{k-1}_{i=0}f((x+i)/k),\quad (k=1,2,3,...) \] of functional equations. We show that any continuous solution of this system is of the form \(f(x)=a\cdot \cot (\pi x)+b\) with some constants a and b. The result contains a new characterization of the cotangent function by means of simple properties. Furthermore we give some information concerning the modified system \[ u(k)f(x)+v(k)=(1/k)\sum^{k- 1}_{i=0}f((x+i)/k),\quad (k\in S), \] which can be used to characterize by the same method Gauss' Psi function and Hurwitz' Zeta function \(x\to \zeta (s,x)\) (Re(s)\(\geq 1\), \(s\neq 1)\).
Mathematics Subject Classification ID: 39B99
Mathematics Subject Classification ID: 33B10
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 3946798
zbMATH Keywords: system
zbMATH Keywords: continuous solution
zbMATH Keywords: cotangent function
zbMATH Keywords: Gauss' Psi function
zbMATH Keywords: Hurwitz' Zeta function
MaRDI profile type: Publication
cites work: Q3889389
cites work: Q5328256
cites work: Elementarer Beweise für die Partialbruchzerlegung des Cotangens
cites work: On (\(1,b_p\))-replicative functions with isolated discontinuities
cites work: Continuous replicative functions
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description: scientific article; zbMATH DE number 3946798