Selected Topics in the Classical Theory of Functions of a Complex Variable

Maurice Heins

• Publisher: Dover Publications
• Year: 2014
• eBook ISBN: 9780486800691
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

Elegant and concise, this text is geared toward advanced undergraduate students acquainted with the theory of functions of a complex variable. The treatment presents such students with a number of important topics from the theory of analytic functions that may be addressed without erecting an elaborate superstructure. These include some of the theory's most celebrated results, which seldom find their way into a first course.
After a series of preliminaries, the text discusses properties of meromorphic functions, the Picard theorem, and harmonic and subharmonic functions. Subsequent topics include applications and the boundary behavior of the Riemann mapping function for simply connected Jordan regions. The book concludes with a helpful Appendix containing information on Lebesgue's theorem and other topics.

[ 1 ]

Preliminaries

1. Terminology, Notation, Chordal Metric, and Topology of the Extended Plane. It will be convenient to list here for future reference some notation and terminology that will be constantly used throughout the present work.

Notation.

Given a set A and E

A, the complement of E with respect to A will be denoted by A − E.

Given

and r a positive real number or + ∞ Δ (a; r) denotes

when a

K, and

when a = ∞. We write (1.2) and (1.3) more briefly as {| z − a| < r} and {| z| > r^-1} respectively and shall follow this practice in describing sets when there is no likelihood of confusion. Similarly, C(a; r) denotes

or

according as a ≠ ∞ or a = ∞. That is, Δ(a; r) is the open circular disk with center a (respectively, ∞) and radius r (respectively, r⁻¹), and, similarly, C(a; r) is the corresponding circumference.

Chordal Metric. In the study of meromorphic functions, whose definition we shall recall explicitly later, it is very convenient for many purposes to employ the chordal metric which was introduced in the theory of functions of a complex variable by Carathéodory (cf. C. Carathéodory, Theory of Functions, chap. 3, vol. 1. New York : Chelsea Publishing Company). We define this as follows. Let S denote the spherical surface in (x₁, x₂, x₃) space with center

and radius

and let π denote the stereographic projection of S onto

, that is, the map that carries the north pole of S into ∞ and a point (x₁, x₂ x₃)

S not the north pole into ξ₁ + iξ₂, where (ξ₁, ξ₂, 0) is the point of intersection of the plane {x₃ = 0} with the line through the north pole of S and (x₁, x₂, x₃). The chordal distance between points a and b of

, denoted by [a, b] is simply the euclidean distance between π⁻¹(a) and π⁻¹(b), π⁻¹ being the inverse of the map π. It is a matter of straightforward computation using the explicit formula for π⁻¹ to verify that

when a and b both belong to K, and that

EXERCISE

1. Verify formulas (1.6) and (1.7).

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Topology o...

Table of contents

• Cover
• Title Page
• Copyright Page
• preface
• content
• Chapter 1. Preliminaries
• Chapter 2. Covering Properties of Meromorphic Functions
• Chapter 3. The Picard Theorem
• Chapter 4. Harmonie and Subharmonic Functions
• Chapter 5. Applications
• Chapter 6. The Boundary Behavior of the Riemann Mapping Function for Simply-connected Jordan Regions
• Appendix
• Index