The continuous, desingularized Newton method for meromorphic functions

H.Th. Jongen, P. Jonker, F. Twilt

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    For any (nonconstant) meromorphic function, we present a real analytic dynamical system, which may be interpreted as an infinitesimal version of Newton's method for finding its zeros. A fairly complete description of the local and global features of the phase portrait of such a system is obtained (especially, if the function behaves not too bizarre at infinity). Moreover, in the case of rational functions, structural stability aspects are studied. For a generic class of rational functions, we give a complete graph-theoretical characterization, resp. classification, of these systems. Finally, we present some results on the asymptotic behaviour of meromorphic functions.
    Original languageEnglish
    Pages (from-to)81-121
    JournalActa applicandae mathematicae
    Issue number1-2
    Publication statusPublished - Sep 1988


    • Asymptotic value
    • Dynamical system
    • Newton method
    • Meromorphic function
    • Phase-portrait
    • Plane (sphere) graph


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