Muir - Complex analysis: a modern first course in function theory

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Library of Congress Cataloging-in-Publication Data:

Muir, Jerry R.

Complex analysis : a modern first course in function theory / Jerry R.

Muir, Jr.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-70522-3 (cloth)

1. Functions of complex variables. 2. Mathematical analysis. I. Title.

QA331.M85 2014

515dc23

2014035668

To Stacey, proofreader, patron, and partner

This unfortunate name, which seems to imply that there is something unreal about these numbers and that they only lead a precarious existence in some people's imagination, has contributed much toward making the whole subject of complex numbers suspect in the eyes of generations of high school students.

Zeev Nehari [21] on the use of the term imaginary number

In the centuries prior to the movement of the 1800s to ensure that mathematical analysis was on solid logical footing, complex numbers, those numbers algebraically generated by adding to the real field, were utilized with increasing frequency as an ever-growing number of mathematicians and physicists saw them as useful tools for solving problems of the time. The 19th century saw the birth of complex analysis, commonly referred to as function theory, as a field of study, and it has since grown into a beautiful and powerful subject.

The functions referred to in the name function theory are primarily analytic functions, and a first course in complex analysis boils down to the study of the complex plane and the unique and often surprising properties of analytic functions. Familiar concepts from calculusthe derivative, the integral, sequences and seriesare ubiquitous in complex analysis, but their manifestations and interrelationships are novel in this setting. It is therefore possible, and arguably preferable, to see these topics addressed in a manner that helps stress these differences, rather than following the same ordering seen in calculus.

This text grew from course notes I developed and tested on many unsuspecting students over several iterations of teaching undergraduate complex analysis at Rose-Hulman Institute of Technology and The University of Scranton. The following characteristics, rooted in my personal biases of how best to think of function theory, are worthy of mention.

• Complex analysis should never be underestimated as simply being calculuswith complex numbers in place of real numbers and is distinguished from being so at every possible opportunity.
• Series are placed front and center and are a constant presence in a number of proofs and definitions. Analyticity is defined using power series to emphasize the difference between analytic functions and the differentiable functions studied in calculus. There is an intuitive symmetry between analyzing zeros using power series and singularities using Laurent series. The early introduction of power series allows the complex exponential and trigonometric functions to be defined as natural extensions of their real counterparts.
• Many properties of analytic functions seem counterintuitive (perhaps unbelievable) to students recently removed from calculus, and seeing these as early as possible emphasizes the distinctive nature of complex analysis. In service of this, Liouville's theorem, factorization using zeros, the open mapping theorem, and the maximum principle are considered prior to the more-involved Cauchy integral theory.
• Analytic function theory is built upon the trinity of power series, the complex derivative, and contour integrals. Consequently, the CauchyRiemann equations, an alternative expression of analyticity tied to differentiability in two real variables, are naturally partnered with the conformal mapping theorem at the end of the line of properties of analytic functions. Harmonic functions, also strongly reliant on this multivariable calculus topic, are the subject of the final chapter, allowing their study to benefit from the full theory of analytic functions.
• The geometric mapping properties of planar functions give intuition that was easily provided by the graphs of functions in calculus and help to tie geometry to function theory. In particular, linear fractional (Mbius) transformations are developed in service to this principle, prior to the introduction of analyticity or conformal mapping.
• The study of any flavor of analysis requires a box of tools containing basic geometric and topological facts and the related properties of sequences. These topics, in the planar setting, are addressed up front for easy reference, so as not to interrupt the subsequent presentation of function theory.

When faced with the choice of glossing over some details to more quickly get to the good stuff or ensuring that the development of topics is logically complete and consistent, I opted for the latter, leaving the readerthe freedom to decide how to approach the text. This was done (with one caveat) subject to the constraint that all material encountered in the typical undergraduate sequence of calculus courses is assumed without proof. This includes results that may not be proved in those courses but whose proofs are part of a standard course in real analysis. This helps to streamline the presentation while reducing overlap with other courses. For example, complex sequences are shown to converge if and only if their real and imaginary components converge. Then the assumed algebraic rules for convergent real sequences imply the same rules for complex sequences. A similar decomposition into real and imaginary parts readily provides familiar rules for derivatives and integrals of complex-valued functions of a real variable. It is important to clarify that material proved in a real analysis course that is not considered in calculus, such as aspects of topology or convergence of sequences of functions, is dealt with here.