Whittaker and Watson

Whittaker and Watson’s analysis textbook is a true classic. My only complaint about the book is that the typesetting is poor. I said years ago that I wish someone would redo the book in LaTeX and touch it up a bit.

I found out while writing my previous post that in fact someone has done just that. That post explains the image on the cover of a reprint of the 4th edition from 1927. There’s now a fifth edition, published last year (2021).

The foreword of the fifth edition begins with this sentence:

There are few books which remain in print and in constant use for over a century; “Whittaker and Watson” belongs to this select group.

That statement is true of books in general, but it’s especially rare for math books to age so well.

The first edition came out in 1902. The book shows its age, for example, by spelling “show” with an e rather than an o. And yet I routinely run into references to the book. Nobody has written a better reference over the last century.

The new edition corrects some errors and adds references for more up-to-date results. But in some sense the mathematics in Whittaker and Watson is finished. This has a bizarre side effect: much of the material in Whittaker and Watson is no longer common knowledge precisely because the content is settled.

The kind of mathematics presented in Whittaker and Watson is very useful, but it falls between two stools. It’s too difficult for undergraduates, and it’s not a hot enough topic of research for graduate students.

When I finished my PhD, I knew some 20th century math and some 18th century math, but there was a lot of useful mathematics developed in the 19th century that I wouldn’t learn until later, the kind of math you find in Whittaker and Watson.

Someone may reasonably object that the emphasis on special functions in classical analysis is inappropriate now that we can easily compute everything numerically. But how are we able to compute things accurately and efficiently? By using libraries developed by people who know about special functions and other 19th century math! I’ve done some of this work, speeding up calculations a couple orders of magnitude on 21st century computers by exploiting arcane theorems developed in the 19th century.

Related posts

Galois diagram

The previous post listed three posts I’d written before about images on the covers of math books. This post is about the image on the first edition of Dummit and Foote’s Abstract Algebra.

Here’s a version of the image on the cover I recreated using LaTeX.

The image on the cover appears on page 495 and represents extension fields. If you’re going through this book sequentially as a text book, it’s likely time will run out before you ever find out what the image on the cover means. If you do get to it, you get to it near the end of your course.

My diagram is topologically equivalent to the original. I took the liberty of moving things around a bit to keep the diagram from being awkwardly wide.

At the bottom of the diagram we have ℚ, the field of rational numbers. At the top of the diagram we have ℚ(i, 21/8), the smallest field containing the rational numbers, the imaginary unit i, and the eighth root of 2. Lines between fields on two levels indicate that the higher is an extension of the lower. The constant ζ in the diagram is

ζ = √i = √2(1 + i)/2.

The significance of the diagram is that extension field relationships like these are important in Galois theory. The image appears late in the book because the majority of the book is leading up to Galois theory.

Book cover posts

When a math book has an intriguing image on the cover, it’s fun to get to the point in the book where the meaning of the image is explained. I have some ideas for book covers I’d like to write about, but here I’d like to point out three such posts I’ve already written.

Weierstrass elliptic function

The book on the left is Abramowitz and Stegun, affectionately known as A&S. I believe the function plotted on the cover is the Weierstrass elliptic function, as I wrote about here.

As the image suggests, my copy of A&S has seen a bit of wear. At one point the cover fell off and I put it back on with packing tape.

Möbius transformation of circles

The book in the middle is my well-worn copy of my undergraduate complex analysis text. The cover is a bit dirty and pages are falling out, a sort of Velveteen rabbit of math books.

I haven’t written a post about the cover per se, but I did write about the necessary math to recreate the image on the cover here. That post explains how to compute the image of a circle under a Möbius transformation. The image on the left is mapped to the image on the right via the function

f(z) = 1/(z – α).

Here α is the point on the right where all the outer circles are tangent. If you wanted to reconstruct the image on the cover, it would be easier to proceed from right to left: start with the image on the right because it’s easier to describe, and apply the inverse transformation using the instructions in the blog post to produce the image on the left.

Hankel functions

I wrote about the book on the right here. I believe the image on the cover is the plot of a Hankel function.

Updates

Here’s a post explaining the image on the cover Abstract Algebra by Dummit and Foote.

And here is a post explaining the image on the cover of A Course of Modern Analysis by Whittaker and Watson.

 

New R book

Five years ago I recommended the book Learning Base R. Here’s the last paragraph of my review:

Now there are more books on R, and some are more approachable to non-statisticians. The most accessible one I’ve seen so far is Learning Base R by Lawrence Leemis. It gets into statistical applications of R—that is ultimately why anyone is interested in R—but it doesn’t start there. The first 40% or so of the book is devoted to basic language features, things you’re supposed to pick up by osmosis from a book focused more on statistics than on R per se. This is the book I wish I could have handed my programmers who had to pick up R.

Now there’s a second edition. The author has gone through the book and made countless changes, many of them small updates that might be unnoticeable. Here are some of the changes that would be more noticeable.

  1. There are 265 new exercises.
  2. Chapter 26 (statistics) and Chapter 28 (packages) got a complete overhaul.
  3. Dozens of new functions are introduced (either in the body of the text or through exercises).
  4. New sections include the switch function (in Chapter 13 on relational operators), algorithm development (in Chapter 23 on iteration), analysis of variance (in Chapter 26 on statistics), and time series analysis (in Chapter 26 on statistics).

I was impressed with the first edition, and the new edition promises to be even better.

The book is available at Barnes & Noble and Amazon.

My densest books

I recently got a copy of Methods of Theoretical Physics by Morse and Feshbach. It’s a dense book, literally and metaphorically. I wondered whether it might be the densest book I own, so I weighed some of my weightier books.

I like big books, I cannot lie.

Morse and Feshbach has density 1.005 g/cm³, denser than water.

Gravitation by Misner, Thorne, and Wheeler is, appropriately, a massive book. It’s my weightiest paperback book, literally and perhaps metaphorically. But it’s not that dense, about 0.66 g/cm³. It would easily float.

The Mathematica Book by Wolfram (4th edition) is about the same weight as Gravitation, but denser, about 0.80 g/cm³. Still, it would float.

Physically Based Rendering by Pharr and Humphreys weighs in at 1.05 g/cm³. Like Morse and Feshbach, it would sink.

But the densest of my books is An Atlas of Functions by Oldham, Myland, and Spanier, coming in at 1.12 g/cm³.

The books that are denser than water were all printed on glossy paper. Apparently matte paper floats and glossy paper sinks.

Volunteer-generated errata pages

I picked up a used copy of Quaternions and Rotation Sequences by Jack B. Kuipers for a project I’m starting to work on. The feedback I’ve seen on the book says it has good content but also has lots of typos. My copy has a fair number of corrections that someone penciled in. Someone on Amazon alluded to an errata page for the book but I’ve been unable to find it.

This made me wonder more generally: Is there a project to create errata pages? I’m thinking especially of mathematical reference books. I’m not concerned with spelling errors and such, but rather errors in equations that could lead to hours of debugging.

I would be willing to curate and host errata pages for a few books I care about, but it would be better if this were its own site, maybe a Wiki.

I don’t want to duplicate someone else’s effort. So if there’s already a site for community-generated errata pages, I could add a little content there. But if there isn’t such a project out there already, maybe someone would like to start one.

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[1] Update: Jan Van lent found the errata page. See the first comment. Apparently the changes that were penciled into my book were copied from the author’s errata list. Also, these changes were applied to the paperback edition of the book.

Books and revealed preferences

Revealed preferences are the preferences we demonstrate by our actions. These may be different from our stated preferences. Even if we’re being candid, we may not be self-aware.

One of the secrets to the success of Google’s PageRank algorithm is that it ranks based on revealed preferences: If someone links to a site, they’re implicitly endorsing it.

I got to thinking about revealed preferences when it comes to reference books the other day when I used some packing tape to keep the cover of my copy of Abramowitz and Stegun from falling off [1].

Instead of asking “What are some of your favorite books,” it might be more informative to ask “Which of your books show the most wear?” [2] This confounds frequent use and poor binding, but that’s life: there are always confounding effects.

My most worn math books are A&S, Bak and Newman, and Dunford and Schwartz .Bak and Newman was my undergraduate complex analysis book; I think it may have had a poor binding. Dunford and Schwartz got a lot of wear in college when I was into functional analysis.

I used A&S a lot in when I was developing a numerical library for Bayesian statistics. I still open it up occasionally, though not as often as I used to.

My volumes of TAOCP are in good shape, but I think that’s because they are well bound. I’ve cracked open Volume 2 quite a bit, though I hardly ever look at the other volumes.

What are some of your most worn books?

Related posts

[1] Yes, I know it’s available online, but I prefer the dead tree edition. And yes, I know there are more extensive references, but in my experience anything I need that isn’t in A&S is unlikely to be in any other reference book.

[2] Benford’s law was discovered via revealed preferences. Simon Newcomb noticed that the early pages of a book of logarithms were much dirtier than the later pages. (Yes, Newcomb discovered Benford’s law, consistent with Stigler’s law of eponymy.)

Banned math book

Courant & Hilbert is a classic applied math textbook, still in print nearly a century after the first edition came out. The actual title of the book is Methods of Mathematical Physics, but everyone calls it Courant & Hilbert after the authors, Richard Courant and David Hilbert. I was surprised to find out recently that this was once a banned book. How could there be anything controversial about a math book? It doesn’t get into any controversial applications of math; it applies math to physics problems, but doesn’t apply the physics to anything in particular.

The book was first published in Germany in 1924 under the title Methoden der mathematischen Physik. Courant says in the preface

… I had been forced to leave Germany and was fortunate and grateful to be given the opportunities open in the United States. During the Second World War the German book became unavailable and was even suppressed by the National Socialist rulers of Germany. The survival of the book was secured when the United States Government seized the copyright and licensed a reprint issued by Interscience Publishers.

Courant’s language is remarkably restrained under the circumstances.

I wondered why the book was banned. Was Courant Jewish? I’d never considered this before, because I couldn’t care less about the ethnicity of authors. Jew or Greek, bond or free, male or female, I just care about their content. The Nazis, however, did care. According to his Wikipedia biography, Courant fled Germany not because of his Jewish ancestry but because of his affiliation with the wrong political party.

***

I never had Courant & Hilbert as a textbook, but I was familiar with it as a student. I vaguely remember that the library copy was in high demand and that I considered buying a copy, though it was too expensive for my means at the time. I recently bought a copy now that the book is cheaper and my means have improved.

I covered most of the material in Courant & Hilbert in graduate school, albeit in a more abstract form. As I mentioned the other day, my education was somewhat top-down; I learned about things first in an abstract setting and got down to particulars later, moving from soft analysis to hard analysis.

One quick anecdote along these lines. I read somewhere that David Hilbert was at a conference where someone referred to a Hilbert space and he asked the speaker what such a thing was. Hilbert’s work had motivated the definition of a Hilbert space, but Mr. Hilbert thought in more concrete terms.

Books you’d like to have read

I asked on Twitter today for books that people would like to have read, but don’t want to put in the time and effort to read.

Here are the responses I got, organized by category.

Literature:

Math, Science, and Software:

History and economics

Religion and philosophy

Misc:

Review of Matrix Mathematics

Bernstein’s Matrix Mathematics is impressive. It’s over 1500 pages and weighs 5.3 pounds (2.4 kg). It’s a reference book, not the kind of book you just sit down to read. (Actually, I have sat down to read parts of it.) I’d used a library copy of the first edition, and so when Princeton University Press offered me a review copy of the second edition, I jumped on it.

Matrix Mathematics has a lot of information on linear algebra. As you’d expect from the title, it’s mostly about linear algebra. And despite it enormous size, it’s also dense. Mostly definitions and theorem statements. Some examples and proofs, but mostly statements of facts.

But there are a lot of other topics covered too. For example, I was surprised to see a section on Bell polynomials, a topic I ran across in my work and blogged about not long ago.

Why even have reference books these days when you can easily find so much online? For one thing, there’s still a lot you can’t easily find online. When you go beyond commonly known material, as this book does, it gets hard to search for what you need.

For another, an author goes to tremendous effort to arrange the information coherently. When you read a book you find things you didn’t know to search for. Maybe you start by looking up what you think you need to know in the index, but then you find out from context what you really needed to know.

I’m glad to add this to the books I keep close at hand. I can find what I need quickly, and that’s more important than it may seem. If I save a couple minutes, the benefit is not just that I get a couple more minutes work done. The main benefit is that I increase my chances of acting on an inspiration before it evaporates.