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Entire Solutions of f(kz) = kf(z)f′(z)

Entire Solutions of f(kz) = kf(z)f′(z) A colleague asked whether x, sin x and sinh x are essentially the only solutions of the non-linear differential equation f(2x) = 2f(x)f′ (x) on [0, +∞) with boundary condition f′(0) = 1. Here we study the formal power series solutions of the more general equation f(kz) = kf(z)f′(z), where k is a non-zero complex number, with f(0) = 0, and we show that a transcendental, formal power series solution exists for only a countable set of k, and that each such solution is an entire function. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Entire Solutions of f(kz) = kf(z)f′(z)

Computational Methods and Function Theory , Volume 12 (1) – Jan 25, 2012

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321827
Publisher site
See Article on Publisher Site

Abstract

A colleague asked whether x, sin x and sinh x are essentially the only solutions of the non-linear differential equation f(2x) = 2f(x)f′ (x) on [0, +∞) with boundary condition f′(0) = 1. Here we study the formal power series solutions of the more general equation f(kz) = kf(z)f′(z), where k is a non-zero complex number, with f(0) = 0, and we show that a transcendental, formal power series solution exists for only a countable set of k, and that each such solution is an entire function.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Jan 25, 2012

References