Holomorphic solutions of an inhomogeneous Cauchy equation
type: Item
instance of: scholarly article
title: Holomorphic solutions of an inhomogeneous Cauchy equation
zbMATH Open document ID: 0679.39002
DOI: 10.1007/BF01836443
author: Luigi Paganoni
author: Stefania Paganoni Marzegalli
published in: Aequationes Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137299
review text: The equation (1) \(\phi (x+y)-\phi (x)-\phi (y)=f(x)f(y)h(x+y)\) is considered, where \(\phi\), f, and h are unknown functions holomorphic in a neighborhood of the origin. The authors find all solutions. The special case of (1) with \(f=h\) had been solved previously by the first author and \textit{I. Fenyö} [Rend. Mat. Appl. VII., Ser. 5(1988), No.3/4, 387-392 (1985; Zbl 0666.39005)]. Normalizations allow one to assume that \(\phi '(0)=0\), and that exactly one of the following four cases holds. A theorem corresponds to each case. (I) \(f(0)=1\), (II) \(f(0)=0\) and \(h(0)=0\), (III) \(f(0)=0\), \(h(0)=1\), and \(h^{(3)}(0)=-f''(0)h''(0),\) and (IV) \(f(0)=0\), \(h(0)=1\), and \(h^{(3)}(0)\neq -f''(0)h''(0).\) For case (I), the solutions are \(f(x)=e^{\sigma x},\) \(h(x)=e^{-\sigma x},\) \(\phi (x)=-1\), for some complex \(\sigma\). The solutions in case (II) are obtained by reducing (1) to the equation solved in the paper cited. In this case, the solution triples (f,h,\(\phi)\) must be of one of three different forms; one of the forms involves the Jacobi elliptic functions while the other two forms involve only elementary functions. The two remaining cases are studied in detail. In each, the solution triples (f,h,\(\phi)\) must be of one of two different forms, one elementary and the other involving elliptic functions.
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 4113187
zbMATH Keywords: holomorphic solutions
zbMATH Keywords: inhomogeneous Cauchy equation
zbMATH Keywords: functional equations in the complex domain
zbMATH Keywords: Jacobi elliptic functions
reviewed by: Francis W. Carroll
MaRDI profile type: Publication
cites work: Q5582452
cites work: Q3695735
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description: scientific article; zbMATH DE number 4113187