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Complex Analysis: In the Spirit of Lipman Bers

Jane P. Gilman, Irwin Kra, and Rubí E. Rodríguez
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 245
[Reviewed by
Allen Stenger
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This is a fairly conventional text for a first course in complex analysis. It is an interesting mix of the concrete and the abstract, and of the formulaic and the geometric. It has good exercises: non-routine and requiring some thought to devise a plan of attack.

It is nominally a graduate text (it is in Springer's series of Graduate Texts in Mathematics ) but there's little in it that would not be accessible to upper-level undergraduates. The most advanced portions deal with function spaces, and it assumes some familiarity with topology. The book covers all the usual topics for a first course and includes a lot of advanced topics such as the Riemann Mapping Theorem, Runge's Theorem on the approximation of analytic functions by rational functions, Moebius transforms, representation of entire functions by infinite products, and a detailed study of the gamma function.

My main gripe about this book is that it gets off to a very slow start. The first half of the book is devoted to proving the equivalence of eight conditions for analyticity. Only after we wade through that do we get any indication of why analyticity is an interesting condition and what you can do with it.

Another book with very similar coverage and treatments but slanted toward the concrete is Bak & Newman's Complex Analysis (Springer, 2nd edition, 1997). In some ways this is a better value — it is the same price but has more pages and goes into more depth, and in particular it has more worked examples. It is in Springer's Undergraduate Texts series, which supports my feeling that Gilman & Kra & Rodriguez is not really a graduate text.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.