International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 633-636

Asymptotic tracts of harmonic functions III

Karl F. Barth1 and David A. Brannan2

1Department of Mathematics, Syracuse University, Syracuse 13244, NY, USA
2The Open University, Department of Pure Mathematics, Milton Keynes MK7 6AA, UK

Received 30 June 1995

Copyright © 1996 Karl F. Barth and David A. Brannan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A tract (or asymptotic tract) of a real function u harmonic and nonconstant in the complex plane 𝒞 is one of the nc components of the set {z:u(z)c}, and the order of a tract is the number of non-homotopic curves from any given point to in the tract. The authors prove that if u(z) is an entire harmonic polynomial of degree n, if the critical points of any of its analytic completions f lie on the level sets τj={z:u(z)=cj}, where 1jp and pn1, and if the total order of all the critical points of f on τj is denoted by σj, then {nc:c}={n+1}{n+1+σj:1jp}.