The index of singular zeros of harmonic mappings of anti-analytic degree one. Issue 1 (2nd January 2021)

• Record Type: Journal Article
• Authors: Luce, Robert; Sète, Olivier
• Is Part Of: Complex variables and elliptic equations. Volume 66:Issue 1(2021)
• Page Start: 1
• Page End: 21
• Publication Date: 2021-01-02
• Subjects: D. Khavinson; Harmonic mappings -- Poincaré index -- singular zero -- multiplicity -- critical set; 31A05 -- 30C55; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2019.1695787
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : We study harmonic mappings of the form f ( z ) = h ( z ) − z ¯, where h is an analytic function. In particular, we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of f, where the Jacobian of f is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on the critical set, where the Jacobian vanishes; such zeros are called singular zeros . Our main result is a characterization of the index of singular zeros, which enables one to determine the index directly from the power series of h .