Periodic Schwarz-Christoffel mappings with multiple boundaries per period

Area of Science:

• Complex Analysis
• Geometric Function Theory
• Conformal Mapping

Background:

• Schwarz-Christoffel (S-C) mappings are fundamental tools for transforming polygonal domains.
• Existing S-C theory primarily addresses simply connected target domains.
• Periodic structures with multiple boundaries per period present challenges for standard S-C methods.

Purpose of the Study:

• To extend the theory of Schwarz-Christoffel mappings.
• To develop new S-C mapping formulas for periodic configurations with multiple polygonal boundaries.
• To address cases where the period window is unbounded or bounded.

Main Methods:

• The study extends S-C mapping theory to periodic domains.
• The preimage domain is a multiply connected circular domain.
• New S-C mapping formulas are derived using the Schottky-Klein prime function.

Main Results:

• Explicit S-C mapping formulas are presented for periodic domains.
• The formulas are expressed using the Schottky-Klein prime function.
• The solution of accessory parameters is discussed, with constructive examples provided.

Conclusions:

• The extended S-C mapping theory provides a constructive method for analyzing periodic structures.
• The developed formulas are applicable to various period window configurations (unbounded or bounded).
• This work offers a significant advancement in the application of conformal mapping to periodic geometric problems.