Length functions of lemniscates
Abstract
We study metric and analytic properties of generalized lemniscates Et(f)=z:ln|f(z)|=t, where f is an analytic function. Our main result states that the length function |Et(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|Et(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.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.