Mathematical Constants II

More than 15 years ago I reviewed Steven Finch's Mathematical Constants for MAA Reviews.  This book was a collection of essays about different numbers that appear in mathematics, ranging from \( \pi \) to the Hardy-Littlewood Constants to Trott's constant 0.1084101512..., which I am sure you all recognize as the unique number whose decimal digits coincide with its partial fraction decomposition.  I wrote in that review that this is the kind of book "to keep in a departmental lounge so that people could flip through it over tea and learn a few fascinating facts," and over the years I have done just that, picking the book up from time to time to learn a curiosity.

We now live in an era where so many canceled tv shows come back for new seasons and movies of our youths get remade that it should not have been a surprise to me when I got the request from MAA Reviews headquarters to take a look at a sequel book, Mathematical Constants II.  Like the best sequels, this one covers similar ground to the original but finds ways to stay fresh and interesting, and my general feelings about this new book are similar to that of the original: I'm not sure who the intended audience or what the intended use for this book is, but any mathematician or math student who picks it up and spends a few minutes with it is likely to find something that is new and of interest to them.  

This volume is divided into five parts.  The first part covers constants in Number Theory and Combinatorics, including constants related to the number of graphs on n vertices satisfying certain properties, the number of integer partitions with various conditions, and various Cohen-Lenstra types of heuristics.  There are sections about average least nonresidues for a given modulus and about the number of Boolean decision functions. This section alone is more than 270 pages long and covers a wide range of mathematics. But subsequent sections cover even more.

The second part of the book is entitled "Inequalities and Approximation", and begins with a discussion of the Hardy-Littlewood Maximal Inequalities from harmonic analysis.  It then moves on to discuss Bessel functions and Airy functions and Bernstein polynomials and many similar functions, mostly motivated by analysis and applied mathematics. Finch is interested in finding, or at least approximating, the zeroes or the limits of these functions and gives a variety of results that help do this.  

A third part covers constants arising in real and complex analysis, including various coefficient estimates for one-to-one analytic functions and results about minimal surface areas for soap bubbles.  This section also includes some problems with exciting names, such as the Belgian Chocolate Problem (what is the smallest positive constant \( \tau \) so that there exists an analytic function with exactly one 0-point at \( z=0 \) and two 1-points at \( z = \tau \) and \( -\tau \)?) and Goddard's Rocket Problem (what thrust function \( u(t) \) would maximize the altitude in the limit?).  

Probability and Stochastic Processes are the topic of the fourth part of the book, which includes sections on Brownian Motion, binary trees, and continued fraction transformations.  The final part deals with constants coming out of Geometry and Topology, with many pages dedicated toward questions such as the expected value of the perimeter of a random triangle in the unit disk or the expected value of the area of a triangle formed by breaking a single line segment in two points chosen uniformly at random.

As you can probably gather, this book covers a lot of ground even for a book that is close to 800 pages in length.  And part of Finch's approach is to keep things brief, not giving many theorems in full or describing the bigger context of why anyone would care about the value of these constants.  However, he does give many references and places that an interested reader can look if they want to know more, and this approach works well. The breadth of topics covered also allows the reader to see how some of the distinctions we make between subfields of mathematics are very loose ones, and that actually these subjects are all interconnected in strange ways.

If I had one quibble with the book it would be that it seems to suffer from a bit of 'mission creep', and many of the sections are not so much about specific constants as they are about sequences or inequalities or the like.  While the first volume contained an index of all of the constants contained within it in numerical order, this one would not be able to do that as many of the topics are not as easily distilled. That said, Finch has once again written a collection of essays about a wide range of topics that I expect I will enjoy flipping through for another decade and a half until I look forward to having Volume III land on my desk.

Darren Glass is the Alumni Professor of Mathematics at Gettysburg College, where his mathematics interests include number theory, graph theory, algebraic geometry, and cryptography.  He is also the Book Reviews editor for the American Mathematical Monthly. He can be reached at dglass@gettysburg.edu.