10 Questions for Mathematician Joel L. Schiff: Author of "Topics in Complex Analysis"

Professor Joel Schiff, University of Auckland, discusses and shares insights on his recently published book Topics in Complex Analysis

1. What inspired you to write this book?

There are some beautiful results in complex function theory not so commonly discussed in other books on the subject. Results such as the Riemann functional equation, the Dirichlet Principle, the 20th century work of Pierre Fatou and Gaston Julia on iterations of rational functions, and that of Nevanlinna on meromorphic functions, among many others. I wanted to make these results more accessible, show that some of them can be considered modern, and discuss advanced topics without making the reader wade through an advanced text for major results.

?2. What do you hope will be the lasting impact of this book?

I hope the reader will come to appreciate that the simple act of the creation of a number of the form ?to solve certain algebraic equations led to a very deep and profound subject that is constantly evolving. As quoted by mathematician Richard Hamming in the frontispiece of the book, “God made the universe out of complex numbers.”

?3.?Can you share any interesting anecdotes from the book??

During the writing of the book, I came across some recent work (2021-2022) that showed that complex numbers are essential in the standard formulation of quantum mechanics. The real-valued standard formalism of QM, which many thought could be a substitute for the complex one, was ruled out. This surprise makes the study of complex analysis all the more compelling.?I briefly noted this in the beginning of the text and immediately notified numerous friends of the result. To study the real world at its most fundamental level (i.e., via quantum mechanics), one must enter the world of complex numbers, the world of the present text.

4. Why is your book important for mathematicians??

Mathematician Paul Erd?s always insisted that God had a book containing the most beautiful and elegant proofs of mathematical theorems. So, I hope that some of my book’s proofs are in the Great Book along with material that would not be so familiar to many analysts. Complex analysis is a major branch of mathematics. Although it exists in a netherworld, it appears in a broad spectrum of scientific fields such as fluid mechanics, signal processing, thermodynamics, mechanical and electrical engineering, quantum mechanics, and so forth. ?Another important factor is the interrelatedness of much of the subject matter. For example, Picard’s Little Theorem has a deep connection to a fundamental theorem in normal families, and the Euler Beta Function is related to the Laplace Transform.

?5. Who is the primary audience for the book??

Anyone with an undergraduate grounding in analysis. Even those working in complex analysis will find topics of interest they had little exposure to. I doubt the ridiculous expression “will be familiar to many readers,[1] but it was attained by Srinivasa Ramanujan (who sent this result to G.H. Hardy) and is in a classic book on String Theory! Indeed, the Euler beta function from 200 years ago gave birth to String Theory. Another interesting notion is that of the M?bius function, a function that on the surface looks useless but appears in a formulation of the Riemann Hypothesis. The notion of M?bius inversion is a powerful tool in physics. The book discusses many of these marvels.

?6. Based on your previous answer to the primary audience of the book, what are the market needs/key challenges this audience faces??

Some topics that would normally require the reader to approach an advanced text on the subject are introduced here for the reader to gain a working knowledge of the topic.

?7. Does your book solve this need/challenge? How??

As one can see from the Table of Contents there are ten topics under consideration. Each is given a substantial treatment with a variety of subtopics.

?8. What unique features do you think make the book stand out??

One would be the inclusion of color images besides the customary line drawings. I am grateful to De Gruyter for allowing me to include these. Demonstrating the steady-state heat flow in a closed region with fixed boundary values could only be shown in this manner. Another would be the treatment of subharmonic and superharmonic functions using their proper definitions without assuming continuity, which is a bit of a cheat. One major theme is the Dirichlet Problem, which is explored from many different perspectives including the Perron method via subharmonic functions.

?Another method of solving the Dirichlet Problem is via a random walk stemming from the work of Kakutani. This is subsequently related to the notion of harmonic measure introduced by Nevanlinna. As an exercise, the reader is invited to write a simple computer program to implement these ideas. A comprehensive treatment of normal families of analytic and meromorphic functions is given. This is both a unifying aspect of various results in function theory and critical in the treatment of the iteration of rational functions initiated by Fatou and Julia. A few of the results are original joint work such as the Arithmetic Fourier Transform which has been employed in signal processing. Other transforms are also presented, such as the Fast Fourier Transform and Laplace Transform. These topics appear in the chapter on Analytic Number Theory where the Riemann Hypothesis and its connection to the Mertens Conjecture (which was proved to be false) is discussed at length.

?9. What inspired you to become a mathematician?

There is something ineffably beautiful in mathematics, which you do not find in other fields. Although scientists in these other fields may disagree, all mathematicians appreciate a beautiful proof when they see one. As a painter I appreciate all kinds of beauty, but mathematics holds a special place. ?An equation such as Euler’s formula is a gift from the gods.

?10.?Are there other books you have written, and on what topics?

Other mathematical books include: Normal Families, The Laplace Transform – Theory and Applications, Cellular Automata – A Discrete View of the World, and The Mathematical Universe – From Pythagoras to Planck. Then there is The Most Interesting Galaxies in the Universe, Rare and Exotic Orchids, and two books on artists, Grace Joel – An Impressionist Portrait, and Marie Bashkirtseff – Portrait of Young Genius.

? [1] In the text it is derived in a specious manner from the Riemann functional equation.