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Publisher:

Imperial College Press

Publication Date:

2006

Number of Pages:

245

Format:

Paperback

Price:

38.00

ISBN:

1860946437

Category:

Textbook

[Reviewed by , on ]

R. B. Burckel

09/11/2006

This is an excellent text for a one-semester course on basic complex analysis for students with modest prerequisite knowledge and mathematical sophistication. In fact, in this category it is the best of many such books the reviewer has seen during his career. (A long out-of-print, comparably good, and even shorter text, by another Englishman coincidentally, was that by G. J. O. Jameson [**MR** 43.3426].)

In a 10-page appendix the author reviews (often with proofs and complete definitions) the needed completeness, compactness and connectedness facts about **R**, but **R** is not *constructed* and **C** is defined (in chapter 1) as **R** × **R** after a preliminary discussion based on the customary *deus ex machina* with *i*. There follows a 16-page chapter on sequences and series in **C** and a 30-page chapter leisurely detailing the needed (elementary) topology of **C**. The remaining 13 chapters (170 pages) cover the basics of complex analysis *in the plane*: analytic functions (defined via **C**-differentiability, Cauchy-Riemann without supplemental hypotheses shown inadequate), exponential and circular functions (defined via power series with Landau's famous definition of π as twice the smallest positive zero of cosine, subsequently shown to coincide with the geometric pi), complex logarithms (after the obligatory obeisance to the tyranny of "multiple-valued" functions, his treatment in actual applications is rigorous and lucid), Cauchy integral theorem (for starlike domains) and formula (for circular contours only, by reduction to the circumferential mean-value theorem via cross-cuts), Morera (with the weak hypothesis ∫_{Δ}*f* = 0 only for triangles, i.e., direct converse of Goursat), Taylor and Laurent series, open-mapping theorem (but, oddly, not the analyticity of *f*^{ -1}), local conformality (geometrically — the Riemann mapping theorem and simple-connectivity are not mentioned), classification of isolated singularities, residues and the argument principle (using — not entirely satisfactorily — local logarithms), maximum modulus principle, Möbius transformations, and harmonic functions (defined via Laplace; no use of Poisson or mention of Dirichlet problem: for real *u* harmonic in a disk the author finds a primitive *f* for the analytic function *D*_{2}*u – iD*_{1}*u* and shows *u*= **Re***f*).

Surprisingly, it is not until the last chapter that uniform convergence is introduced (but there both Weierstrass' and Hurwitz's theorems are proved). In those places where one would expect it to be used (e.g., power series), it is circumvented by careful estimates of the remainders, which probably has some pedagogical advantages. Generally the author has carefully decided what material is essential in such a course and crafted his tools to those ends, never over or under equipping his readers.

Some (other) outstanding features: the text is practically a *Taschenbuch*, but the typography is uncrowded and pleasing. Despite being detailed and meticulous (even linguistically — we see "alternative" proofs, not "alternate" proofs) the exposition is neither oppressive nor pedantic, in fact almost leisurely. Many sections receive succinct verbal summaries under the rubic "What's going on?".

After these ecomia, two defects should be noted: The index is inadequate and (cardinal sin in a mathematics book and despair of the non-linear reader) there is * no* symbol index. Also, no exercises.

R. B. Burckel is Professor of Mathematics at Kansas State University. He is the author of many books and papers on classical complex analysis. He also translated the superlative textbook by a "grand old man" of the subject, Reinhold Remmert.

- Complex numbers
- Sequences and Series
- Metric Space Properties of the Complex Plane
- Analytic Functions
- The Complex Exponential and Trigonometric Functions
- The Complex Logarithm
- Complex Integration
- Cauchy’s Theorem
- The Laurent Expansion
- Singularities and Meromorphic Functions
- Theory of Residues
- The Argument Principle
- Maximum Modulus Principle
- Möbius Transformations
- Harmonic Functions
- Local Properties of Analytic Functions

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