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The Calculus of Complex Functions

William Johnston
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AMS/MAA Textbooks
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
John Ross
, on
In The Calculus of Complex Functions, William Johnston offers an introduction to complex analysis that emphasizes and centers calculus. The text serves to “describe complex analysis as a natural augmentation of the calculus on real functions.” It does this by immediately introducing the Extension Theorem at the start of the text (the Extension Theorem being the result by which a real analytic function can extend its domain into the complex plane, upon which it is a holomorphic function). By introducing this result immediately, Johnston is able to center analytic (real and complex) functions early in the text. The result is a more challenging, but – for the right audience – a more rewarding introduction to Complex Analysis, reaching vistas often not seen until graduate school.
This is a well-organized text. There are six chapters, structured so that the first three (Analytic Functions and the Derivative; Complex Integration; and Non-Entire Functions) are largely sequential, while the final three (Solving the Dirichlet Problem; Further Topics and Famous Discoveries; and Linear Algebra and Operator Theory) represent advanced topics that are written to be largely independent of each other. This allows one to tailor a semester-long class to (for example) cover the first three chapters, plus a fourth chapter. The topics covered in the final three chapters are truly advanced, and many students would not see these topics otherwise until the graduate level.
In addition to being well organized, this is a well written and beautifully illustrated text. (More on the beautiful images later.) The narrative voice is precise and pays careful attention to the analytic details, but remains somewhat informal and steers away from being too dry. There are a number of features that embellish the text, including embedded Explorations, end-of-section Important Concepts, and end-of-chapter Historical Notes. All of these are appreciated – the historical notes are a particularly nice touch. Scattered throughout the text are Engaged Learning Modules, which offer explorations in Mathematica or in external sites and applets. Each section concludes with an assortment of exercises that seem appropriately challenging.
The images of complex functions in the text are of particular note and warrant their own separate consideration. Functions are graphed using Mathematica, and are true 4-D renderings of the graph in question. The domain continues to be graphed on the Cartesian plane, but functions’ output values use height and color to represent the output’s modulus and argument, respectively.
Who would get the most out of this text? I believe there is a clear case to be made that an advanced undergraduate (or graduate) student, grounded with a strong background in Real Analysis and proof-writing, would gain a lot from using this text. The level of care and precise detail offered to the proofs in this text is exemplary – they are easy to read and easy to follow. Additionally, the text goes further and offers deeper results than many other texts of its kind – the advanced undergraduate student who tackles the later chapters would be able to engage with graduate-level topics and glimpse research-level questions. At the same time, I do think that this text would be challenging for a less advanced student. Johnston argues in his preface that “teachers can direct students at different levels of theoretical engagement, from gainfully digesting every detail to merely skimming important proofs, just as they can when teaching calculus that offers Cauchy’s delta-epsilon description of limits or a proof of the product rule.” I admit I am a little skeptical of this claim – this text has theory and proofs at its heart, especially in the first few sections as it lays a strong groundwork for the Extension Theorem. I suspect a teacher who encouraged their students to simply skim the proofs in this book would find their class a bit adrift in this early chapter, and I would be hesitant to introduce it to a student who had taken (for example) Calculus 3 but no course in proof-writing.
Overall, The Calculus of Complex Functions is a very clear, yet very advanced, introduction to its subject matter, with great care given to the proofs, the narrative voice, and the Mathematica images. The images alone are well worth the price of admission --- and a student with a solid understanding of real analysis will gain a great deal from the vistas this book offers.
John Ross is an assistant professor of mathematics at Southwestern University.