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What are your favourite scholarly books? My favourite is definitely G.N. Watson's "A treatise on the theory of Bessel functions" (full text). Every single page is full of extremely precise references, ranging over 300 years of mathematics and hundreds of papers. One gets the definite impression that Watson carefully studied each and every paper he refers to.

In an entirely different field, I would say that Umberto Eco's "The search for the perfect language" (about linguistics and the evolution of language) is similarly scholarly.

Note that this question is focused on books displaying great depth and breadth of knowledge of the relevant literature, not on great mathematical writing or ``good'' undergraduate level math books (though there is not necessarily an empty intersection).

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I think this question is a little too off-topic; it would make a better blog post than an MO question. –  Qiaochu Yuan Feb 28 '10 at 22:30
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Maybe "scholarly" is a better word than "erudite" here? I have almost always seen "erudite" used to describe a person rather than a work. Just a suggestion... –  Pete L. Clark Feb 28 '10 at 22:32
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This should most definitely be community wiki! –  Grétar Amazeen Feb 28 '10 at 22:42
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Yes, Umberto Eco's book displays fantastic erudition and yet is quite amusing. I loved Jan van Gorp's claim that the original language spoken in Päradise by Adam (with Eve, I presume?) was Dutch, and more precisely the dialect of Antwerp... –  Georges Elencwajg Feb 28 '10 at 23:16
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I've wiki-hammered this thread. I agree with Qiaochu that this seems pretty off topic. Without some explanation of exactly what you're trying to get out of a list of books, I think you'll just end up with a long list that nobody ever reads. For what it's worth, I now think the two posts you linked to also lacked focus, but those were the early days of MO when we were figuring things out. Could you at least make it clearer why this is not a duplicate of those questions? Are you asking for favorite books that aren't great writing? Is there an example of a book that belongs here but not there? –  Anton Geraschenko Mar 1 '10 at 0:19
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19 Answers

Donald Knuth's series of books The Art of Computer Programming. He even credits the source of the exercises at the end of chapters if he found them elsewhere.

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I strongly support that, every sentence feels like it could easily grow into a treatise in its own right. –  Edmund Harriss Oct 14 '11 at 4:41
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H.S.M. Coxeter's Regular polytopes. Opening it in a random page and rereading makes me want to hug him.

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I think you mean Coxeter's Regular Polytopes, in which case I concur. –  John Stillwell Mar 1 '10 at 2:55
    
Sure I did! :) –  Mariano Suárez-Alvarez Mar 1 '10 at 2:58
    
I read the post and was going to suggest it. Beautiful book, it's a large part of what got me into math in the first place! –  Charles Siegel Mar 1 '10 at 3:01
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Perhaps the archetypical example of a scholarly work in mathematics is L.E. Dickson's three volume History of the Theory of Numbers. The aforelinked wikipedia page puts it rather well:

"The 3-volume History of the Theory of Numbers (1919–23) is still much consulted today, covering divisibility and primality, Diophantine analysis, and quadratic and higher forms. The work contains little interpretation and makes no attempt to contextualize the results being described, yet it contains essentially every significant number theoretic idea from the dawn of mathematics up to the 1920s. A planned fourth volume was never written. A. A. Albert remarked that this three volume work 'would be a life's work by itself for a more ordinary man.'"

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Conway and Sloane: Sphere Packings, Lattices and Groups. I've only flipped through it but that was enough to convince me that it is an example of what the OP is looking for. The first chapter, in particular, seems to be almost all citations.

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"Development of mathematics in the 19th century" by Felix Klein is a great book. Is it a good example here?

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I am very fond of Narkiewicz's Elementary and Analytic Theory of Algebraic Numbers. The bibliography is more than 170 pages long. The end of the chapter notes are wonderful. I find Chapter 1 to be such a marvel of scholarly exposition that I keep reading it over and over again -- it's hard for me to go on to Chapter 2!

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Peter Johnstone's "Sketches of an Elephant" - the Topos Theory compendium.

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I vote for Titchmarsh's "The Theory of the Riemann Zeta function." Very complete (for its time) and lucid.

I would also add "A Panoramic View of Riemannian Geometry" by Marcel Berger.

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A rather amazing book, in the spirit of Dickson's on the history of number theory, is Th. Muir's The Theory of Determinants in the Historical Order of Development which, in four volumes, gives more information and references about determinants than most humans can survive.

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Yes, Mariano, it's one of the very best: I thought of it within a millisecond of seeing the question. –  Victor Protsak Sep 16 '10 at 1:51
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Éléments de mathématique by Nicolas Bourbaki.

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Nelson Dunford and Jacob T. Schwartz's Linear Operators. One of the most awe-inspiring books ever written.

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I think Functional Analysis by Riesz-Nagy fits the criteria quite well; there are copious footnotes (I read it a while ago and that is probably the part of it that I remember best!).

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Anything by John Milnor. His little book Morse Theory is a very clear, concise introduction to certain essential aspects of differential topology and Riemannian geometry, starting at a fairly elementary level and winding up with Bott periodicity for unitary groups. In this vein, his Characteristic Classes is similarly clear and concise. To my mind, Milnor is an extremely gifted expositor. From a more scholarly point of view, Kobiyashi and Nomizu's two volumes on differential geometry (can't recall the exact title right now) are pretty comprehensive, both in material covered and in references. And since theoretical physics is within the purview of MO, I think The Feynman Lectures on Physics, vols. I,II,III are a work of real genius. When I was an undergrad at Caltech from 1968-72, we used them for introductory physics; students jokingly called them "the big red sleeping pills" because the material went down so easily it might make one doze off. His Quantum Electrodynamics provides a beutifully intuitive introduction to a fairly abstruse subject. Finally, another of may favorites is Abraham and Marsden's Foundations of Mechanics, both in terms of exposition and scholarship.

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I love Feynman's QED. You realize what he's doing with the arrows in the first chapter (I still remember that little, "Oh! They're just complex numbers for physicists who are scared of math!" in my head!) and then everything becomes so straightforward. I hope that by the time I could write a book on something that my thought processes are that intuitively clear. –  drvitek Sep 15 '10 at 22:08
    
Oh yeah, another thing about "The Big Red Sleeping Pills": you couldn't remember what you thought you learned the next morning; it went down that easily. –  drbobmeister Dec 2 '10 at 0:23
    
But you really had to be there for the homework. Principle of Virtual Work in the first week of Physics 1. Help! –  drbobmeister Dec 2 '10 at 0:24
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Abramowitz and Stegun's Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, and its successor, the NIST Handbook of Mathematical Functions.

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Pretty much anything by Serge Lang, as far as I can tell. For instance, his Differential and Riemannian Manifolds contains asides (especially in the preface) with references to all sorts of papers and other books in differential geometry and topology---which I find all the more remarkable since Lang was a number theorist.

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@AM: Lang was a mathematician! :) How do you think he acquired his expertise in subjects outside of his core knowledge? By reading the literature. I was tempted to mention Lang's Algebra. It's true that Lang was sometimes a little sloppy: not everything he writes in his books is literally correct, which is often regarded as a failure of scholarship. I personally believe that these imperfections have been exaggerated by about two orders of magnitude, and overall he is more reliable in the details (and unquestionably, in the spirit and insight) than most other authors. –  Pete L. Clark Feb 28 '10 at 23:32
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Of course, but I just tend to be in awe of people like Lang (and others such as Terence Tao, Jean-Pierre Serre, etc.) who manage to be that "well-rounded." –  Akhil Mathew Feb 28 '10 at 23:56
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Naimark's Normed rings has something of the scholarly flavour being discussed. It is extremely complete in its exposition, with extensive references and notes, and many asides and elabolrations which (at least in the English translation that I studied) were typeset in a very small font (the kind that you might use for notes added in proof). Nevertheless, the book was far from short (and grew with subsequent enditions, I think).

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"The Riemann legacy: Riemannian ideas in mathematics and physics" By Krzysztof Maurin

I recommended it already in my answer to the following MO question:

Historical basis and mathematical significance of Riemann surfaces

But there is much more in there than Riemann surfaces:

http://books.google.com/books?id=jlll448aDLEC&printsec=frontcover&dq=inauthor:Maurin&hl=en&ei=5CpyTr2-HOGusQLfz-X1CQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CDUQ6AEwAg#v=onepage&q&f=false

As for Umberto Eco's books, my favorite is:

Kant and the Platypus : Essays on Language and Cognition (ISBN 0-15-601159-X) :)

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I just found a 2007 PhD thesis on topics surrounding intuitionistic type type theory that has blown me away.

Warning: many of the regulars on MO are likely to find this work way too philosophical for their tastes (as per their many comments on MO). They should also note that his copious references heavily favour slightly older mathematics, but by mathematicians of high stature. In other words, as I have said here before, it really seems like "mathematical taste" has shifted far away from the more philosophical stance that some of the Giants of mathematics used routinely.

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