Abstract
The classical Schwarz‐Christoffel formula gives conformal mappings of the upper half‐plane onto domains whose boundaries consist of a finite number of line segments. In this paper, we explore extensions to boundary curves which in one sense or another are made up of infinitely many line segments, with specific attention to the “infinite staircase” and to the Koch snowflake, for both of which we develop explicit formulas for the mapping function and explain how one can use standard mathematical software to generate corresponding graphics. We also discuss a number of open questions suggested by these considerations, some of which are related to differentials on hyperelliptic surfaces of infinite genus.
Highlights
Let D be a polygon in the complex plane with n vertices and interior angles παi, 0 ≤ αi ≤ 2, 1 ≤ i ≤ n; the exterior angles are given by πμi, where αi μi 1
The Schwarz-Christoffel mapping of the upper-half plane onto D is effected by a function of the form fz α z dx 0 x − ai μ1 · · · x − an μn β, 1.1 where a1 < · · · < an are real and α, β ∈ C
The classical Schwarz-Christoffel formula can be applied to each finite product on the right-hand side, which gives a staircase with a finite number of steps
Summary
Let D be a polygon in the complex plane with n vertices and interior angles παi, 0 ≤ αi ≤ 2, 1 ≤ i ≤ n; the exterior angles are given by πμi, where αi μi 1. Even though 1.1 gives the appearance of being an explicit formula, there is no known relation between the values of the ai and the lengths of the sides of D. For this reason, when considering an infinite-sided polygon, one needs to justify passage to a corresponding infinite product, which is what we do in what follows. We explore the case in which there are infinitely many points ai on the real line or the unit circle and obtain results for two kinds of “polygons.”
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