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Length functions of lemniscates

Olga Kuznetsova·Vladimir Tkachev
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Abstract

We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulas of the length function in some special cases.

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Olga Kuznetsova, Vladimir Tkachev. 2009-03-01. Length functions of lemniscates. https://doi.org/10.1007/s00229-003-0411-3

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