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Conformal Mappings of Simply Connected Domains

In: Lectures on Complex Analysis

Author

Listed:
  • Norbert Steinmetz

    (TU Dortmund, Department of Mathematics)

Abstract

This chapter focuses on the Riemann Mapping Theorem. Building on the Schwarz reflection principle, which is of independent interest, we discuss conformal mappings of domains bounded by rectilinear and circular polygons, and conic sections. Special applications include conformal mappings of doubly connected domains onto annuli, the geometric definition of the elliptic modular function, and an analysis of the boundary behaviour of proper mappings for multiply connected domains. The final section is devoted to the class $$\mathcal S$$ S of normalized conformal mappings of the unit disc.

Suggested Citation

  • Norbert Steinmetz, 2026. "Conformal Mappings of Simply Connected Domains," Springer Books, in: Lectures on Complex Analysis, chapter 8, pages 183-214, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-13996-2_8
    DOI: 10.1007/978-3-032-13996-2_8
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