Geometric Function Theory : A Second Course in Complex Analysis / by Tom Carroll.

Format:

Book

Author/Creator:

Carroll, Tom.

Series:

Springer Undergraduate Mathematics Series, 2197-4144

Language:

English

Subjects (All):

Functions of complex variables.

Geometry, Hyperbolic.

Functions of a Complex Variable.

Hyperbolic Geometry.

Local Subjects:

Functions of a Complex Variable.

Hyperbolic Geometry.

Physical Description:

1 online resource (358 pages)

Edition:

1st ed. 2024.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2024

Summary:

This textbook provides a second course in complex analysis with a focus on geometric aspects. It covers topics such as the spherical geometry of the extended complex plane, the hyperbolic geometry of the Poincaré disk, conformal mappings, the Riemann Mapping Theorem and uniformisation of planar domains, characterisations of simply connected domains, the convergence of Riemann maps in terms of Carathéodory convergence of the image domains, normal families and Picard's theorems on value distribution, as well as the fundamentals of univalent function theory. Throughout the text, the synergy between analysis and geometry is emphasised, with proofs chosen for their directness. The textbook is self-contained, requiring only a first undergraduate course in complex analysis. The minimal topology needed is introduced as necessary. While primarily aimed at upper-level undergraduates, the book also serves as a concise reference for graduates working in complex analysis.

Contents:

1 Introduction

2 The Complex Plane - Preparatory Topics

3 The Riemann Sphere

4 The Hyperbolic Disk

5 Normal Families and Value Distribution

6 Simply Connected Domains and the Riemann Mapping Theorem

7 Runge's Theorem and Further Characterisations of Simply Connected Domains

8 Univalent Functions - the Basics

9 Carathéodory Convergence of Domains and Hyperbolic Geodesics

10 Uniformisation of Planar Domains.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Carroll, Tom Geometric Function Theory

ISBN:

9783031737275

OCLC:

1481785586