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Nine Introductions in Complex Analysis

Sanford L. Segal
Publication Date: 
Number of Pages: 
North-Holland Mathematics Studies 208
[Reviewed by
John D. Cook
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In the preface to Nine Introductions in Complex Analysis (2nd edition), author Sanford Segal explains the motivation for his book. While there is general agreement as to what a first course in complex analysis should contain — power series, the Cauchy integral formula, contour integration, etc. — there is no such consensus regarding the syllabus for a second course. To this end, Segal covers nine topics that might reasonably be part of a second course in complex analysis. (Covering all nine in one course would be too much. But the topics are largely independent and so one could select some subset of the topics.)

Nine Introductions intersperses formal theorems and proofs with numbered sections it calls "notes." These notes provide informal commentary, historical remarks, etc. The coverage of Jensen's formula illustrates the contrast between the formality of the theorems and the informality of the notes. Jensen's formula is stated as follows.

Suppose f is analytic on B(0, R) and f(0) ≠ 0. Let r1, r2, ... be the moduli of the zeros of f in B(0, R) arranged in a non-decreasing sequence. Then if rn < rrn+1,

log frac{r_n}{r_1 ldots r_n} = frac{1}{2pi} int_0^{2pi} log | f(r e^{i	heta}) |, d	heta - log|f(0) |.


Three notes follow the statement of the theorem. The second explains:

Jensen's formula may be interpreted as saying roughly that the more zeros an entire function f(z) has, the faster it must grow as |z| → ∞ (the converse of this idea is obviously false as iterated exponentials show).

The notes are one of the best features of the book. The notes include the kinds of comments that mathematicians often verbalize in lectures but are reluctant to state in print.

Some chapters center around theorems that are at least easy to state, even if they may be difficult to prove: the Riemann mapping theorem, Picard's theorems, etc. The extreme example is the chapter on the Riemann hypothesis; the hypothesis remains unproven, though it is not difficult to state.

Other chapters center around more technical developments. Many of the theorems in these chapters are lengthy to state and difficult to interpret. I would like to have seen more expository comments to explain the problems that motivated the abstruse theorems. But even in the more technical sections, there are occasional theorems that are simple to state. The best example may be a theorem by Nevanlinna. In the midst of a highly technical chapter, the following theorem stands out for its simplicity and surprising conclusion.

Suppose f(z) and g(z) are two functions meromorphic in the plane. Suppose also that there are five distinct numbers a1, …, a5 such that the solution sets {z: f(z) = ai} and {z: g(z) = ai} are equal. Then either f(z) and g(z) are equal everywhere or they are both constant.

One chapter of Nine Introductions is devoted to elliptic functions. The study of elliptic functions can be confusing because the subject can be approached from seemingly unrelated directions. The book gives a very clear introduction to elliptic functions, demonstrating the connections between various perspectives. The stress on connecting multiple perspectives runs throughout the book; it is perhaps most pronounced when covering elliptic functions. The book lives up to its promise in the foreword that

... multiple proofs of the same major result are frequently given in the belief that the demonstration of different points of view can only serve to elucidate a subject.

As George Pólya once said,

It is better to solve one problem five different ways, than to solve five problems one way.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.
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