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Explorations in Complex Analysis

Explorations in Complex Analysis

Michael A. Brilleslyper
Michael J. Dorff
Jane M. McDougall
James S. Rolf
Lisbeth E. Schaubroeck
Richard L. Stankewitz
Kenneth Stephenson
Copyright Date: 2012
Edition: 1
Pages: 392
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  • Book Info
    Explorations in Complex Analysis
    Book Description:

    This book is written for mathematics students who have encountered basic complex analysis and want to explore more advanced project and/or research topics. It could be used as (a) a supplement for a standard undergraduate complex analysis course, allowing students in groups or as individuals to explore advanced topics, (b) a project resource for a senior capstone course for mathematics majors, (c) a guide for an advanced student or a small group of students to independently choose and explore an undergraduate research topic, or (d) a portal for the mathematically curious, a hands-on introduction to the beauties of complex analysis. Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on student-driven and technology-enhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and hands-on manipulation. There are more than 15 Java applets that allow students to explore the research topics without the need for purchasing additional software.

    eISBN: 978-1-61444-108-3
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. Introduction
    (pp. xi-xviii)

    This book is written for undergraduate students who have studied some complex analysis and want to explore additional topics in the field. It could be used as

    a supplement for an undergraduate complex analysis course allowing students to explore a research topic;

    a guide for undergraduate research projects for advanced students;

    a resource for senior capstone courses; or

    a portal for the mathematically curious, a hands-on introduction to the beauties of complex analysis.

    This book differs from other mathematics texts. It focuses on discovery, self-driven investigation, and creative problem posing. The goal is to inspire students to investigate, explore, form...

  4. 1 Complex Dynamics: Chaos, Fractals, the Mandelbrot Set, and More
    (pp. 1-84)
    Richard L. Stankewitz and James S. Rolf

    This chapter introducescomplex dynamics, an area of mathematics that has inspired and continues to inspire research and experimentation. Its goal is not to give a comprehensive description of the topic, but to engage you with the general notions, questions, and techniques of the area and to encourage you to actively pose as well as pursue your own questions.

    Dynamics, in general, is the study of mathematical systems that change over time, i.e.,dynamicalsystems. For example, consider a Newtonian model for the motion of the planets in our solar system. Here the mathematical system is a collection of variables...

  5. 2 Soap Films, Differential Geometry, and Minimal Surfaces
    (pp. 85-160)
    Michael J. Dorff and James S. Rolf

    Minimal surfaces are beautiful geometric objects with interesting properties that can be studied with the help of computers. Some standard examples of minimal surfaces in R³ are the plane, Enneper’s surface, the catenoid, the helicoid, and Scherk’s doubly periodic surface (see Figure 2.1; the images shown are part of the surfaces, which continue on). Minimal surfaces are related to soap films that result when a wire frame is dipped in soap solution. To get a sense of this connection, consider the following problem.

    Steiner Problem: Four houses are located so that they form the vertices of a square that has...

  6. 3 Applications to Flow Problems
    (pp. 161-196)
    Michael A. Brilleslyper and James S. Rolf

    This chapter developed from a series of lectures prepared for an undergraduate mathematical physics course. The lectures were designed to show students applications of complex function theory that connected to familiar topics from calculus and physics. Two dimensional flows of ideal fluids was a natural topic. Many ideas from vector calculus are used and there are numerous applications of the methods that are developed. Modeling ideal fluid flow is a standard application of conformal mappings and is readily found in most undergraduate complex analysis texts (see [5] or [1]). However, in preparing the notes it became apparent that there was...

  7. 4 Anamorphosis, Mapping Problems, and Harmonic Univalent Functions
    (pp. 197-270)
    Michael J. Dorff and James S. Rolf

    Complex-valued analytic functions have many nice properties that are not possessed by real-valued functions. For example, we say a complex-valued function is analytic if we can differentiate it one time. It is true that if a complex-valued functionfis analytic, then we can differentiate it infinitely many times. Complex-valued analytic functions can always be represented as a Taylor series, and they are conformal (that is, they preserve angles when$f' \ne 0$), properties that are not true for real-valued functions that can be differentiated one time. Why does an analytic function have these properties? Iff=u+ivis...

  8. 5 Mappings to Polygonal Domains
    (pp. 271-316)
    Jane M. McDougall, Lisbeth E. Schaubroeck and James S. Rolf

    A rich source of problems in analysis is determining when, and how, we can create a one-to-one function of a particular type from one region onto another. In this chapter, we consider the problem of mapping the unit disk D onto a polygonal domain by two different classes of functions. For analytic functions we give an overview and examples of the well known Schwarz-Christoffel transformation. We then diverge from analytic function theory and consider the Poisson integral formula to find harmonic functions that will serve as mapping functions onto polygonal domains. Proving that harmonic functions are univalent requires us to...

  9. 6 Circle Packing
    (pp. 317-344)
    Kenneth Stephenson

    Complex analysis is in many ways the ultimate in continuous mathematics. It presents you with a smooth world: continuous variables, infinitely differentiable functions, smooth surfaces, backed up by a complex arithmetic with its power series, line integrals, and all sorts of handy formulas. In this chapter, however, we will develop a different view of the topic. Here the geometry behind analytic functions moves strongly to the fore as we see how one might discretize complex analysis.

    An analogy is that of a mountain stream. We normally treat the stream as a continuous medium and use continuous variables and functions to...

  10. A Background
    (pp. 345-356)
  11. B The Riemann Sphere
    (pp. 357-364)
  12. Index
    (pp. 365-370)
  13. About the Authors
    (pp. 371-373)