ASYMPTOTIC TRACTS OF HARMONIC FUNCTIONS III

A tract (or asymptotic tract) of a real function u harmonic and nonconstant in the complex plane C is one of the n_c 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 ha...

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Bibliographic Details
Published in:International Journal of Mathematics and Mathematical Sciences 1996, Vol.1996 (4), p.633-636
Main Authors: Barth, Karl F., Brannan, David A.
Format: Article
Language:English
Subjects:
Asymptotic tracts
harmonic functions
Online Access:Get full text
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Summary:A tract (or asymptotic tract) of a real function u harmonic and nonconstant in the complex plane C is one of the n_c 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 τ_□={z:u(z)=c_□}, where 1≤ □ ≤p and p ≤ n-1, and if the total order of all the critical points of f on τ_□ is denoted by σ_j, then (The equation is abbreviated).
ISSN:0161-1712
1687-0425
DOI:10.1155/S0161171296000890