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The arc length of the lemniscate \(| w^2+c|=1\). (Chinese. English summary) Zbl 0926.31001

Summary: Let \(s(c)\) be the arc length of the lemniscate \(| w^2+c|=1\), \(c\in [0,+\infty)\). The author discusses some properties of the function \(s(c)\), and solves the case \(n=2\) of a conjecture proposed by P. Erdős, F. Herzog and B. Piranian [J. Anal. Math. 6, 125–148 (1958; Zbl 0088.25302)].

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30B10 Power series (including lacunary series) in one complex variable
30C85 Capacity and harmonic measure in the complex plane
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