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Complex Analysis

John M. Howie
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Michael Berg
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For better or for worse, I am becoming more and more of an anglophile as I read and study more and more papers and books by mathematical authors from other national origins. It's a bit of a wild opinion, I know, and probably politically incorrect (whence an even more attractive proposition) to hold that there is such a thing as a national character and a corresponding national style. Nonetheless, there is strong evidence that such enviable stylistic elements as clarity and elegance of expression are somehow more common among mathematical authors from the British Isles (but Jean-Pierre Serre is the great Gegenbeispiel to my thesis, of course). In my own student days, now long ago, I recall being enthralled by Burkill's book Lebesgue integration, by Hardy and Wright (i.e. their classic Introduction to the Theory of Numbers), and by Titchmarsh on analytic functions. There was something very special about the way these authors expressed themselves, conveying emotions somehow, even as they headed straight for the heart of the subjects under consideration. Prudence dictates that I desist from naming examples "from the other side," i.e. standard sources which despite their importance are all but unreadable. (I'm sure we all have our own candidates.)

It is with this perhaps some what controversial axiom posited that I now come to John M. Howie's Complex Analysis.

Well, it's more of the same! Howie's book is a gem. I want to use it the next time I teach complex analysis. Not only do Howie's selection of topics and their sequence correspond perfectly to what I believe to be the ideal approach to this gorgeous subject, the writing style is (again) wonderful. Consider the following sample: "Since we shall require Cauchy's Theorem and its consequences for contours that are neither convex nor polygonal, it becomes a duty on the author's part to present a proof of a more general case. Whether there is a corresponding duty on the reader's part is left to individual conscience! There is no doubt, however, that useful skills follow from the mastery of substantial proofs." All in all a wonderful example of sound pedagogy by merely dropping the right hint. So many contemporary texts quickly embrace condescension and proceed mainly to annoy the reader.

As regards technical points, the book is split into twelve chapters, each of which is split into a relatively small number of short and sweet sub-sections which can be easily used to build individual lectures. It's nigh on a perfect text-book in this way. There are also a number of wonderful ideological passages; see e.g. 2.1 "Are complex numbers necessary?" and 3.1 "Why is complex analysis possible?" Beyond this there are "the right number of exercises" for complex analysis: not too many, not too few, a huge number of proofs in them, but also such things as descriptions of multi-valued functions, determinations of specific Möbius transformations, etc., etc. And here is problem 4.18

Comment, on the mathematical rather than the literary content, of

Little Jack Horner sat in a corner
Trying to work out π.
He said, "It's the principal logarithm
Of (-1)-i."

It doesn't get much better than that!

So, clearly, I think this is a terrific book. I'm going to use it the first chance I get. And I recommend it very, very highly.

Michael Berg ( is professor of mathematics at Loyola Marymount University.