This is an excellent text that could be used to support several second and higher-year graduate courses in real variables and number theory. The book also has value in upper level undergraduate courses for students who have taken a basic complex variable course.

Written by authors who are experts in their fields, the book contains roughly two dozen sections spanning seven basic mathematical areas: algebra, approximation, operator theory, harmonic analysis, Banach algebras, complex dynamics and number theory. Each section illustrates how using complex variables can lead to a shorter, punchier, more elegant proof. The authors’ motivation for writing the book was the observation that use of complex variables is not as emphasized nowadays as it was in the first half of the 20th century.

To fully appreciate the book’s theme, let us briefly review the difference between proofs in undergraduate versus graduate courses. Proofs and proof exercises in early year undergraduate courses may be a) line by line proofs, for example, the proof that the identity of a group is unique, b) proofs based on induction, or c) proofs based on simple estimates such as seen in a variety of places in Calculus 1–3. More advanced proofs, such as those seen in the Junior, Senior and 1st year graduate courses may focus on proof techniques specific to certain subject matter, for example, the importance of short exact sequences or structure theorems in algebra or the use of Taylor’s theorem with remainder in analytic courses such as calculus or probability.

When students get to higher level graduate courses — roughly the second and higher years — they should routinely be exposed to multiple proof methods, to the idea that the same theorem may have many proofs, depending on the approach used. It is this point which the book addresses.

**Example 1:** As a simple example, one can teach a course in real Fourier analysis and prove the fundamental Fourier uniqueness theorem using real methods. As *Complex Proofs* charmingly shows (in section 4.1), one can alternatively use a complex variable approach resulting in a shorter punchier proof.

**Example 2:** In a number theory course, one can approach the prime number theorem — the assertion that the number of primes less than *x* is asymptotically *x**/*log(*x*) as *x* tends to infinity — by different methods and approaches. Using discrete methods one can define special functions and get good upper and lower bounds. Alternatively however, one can use a complex variables approach, an approach that yields the entire proof. A good number theory course, and indeed a good number theory text, should expose students to both methods. Although I am familiar with the complex proof of the prime number theorem I was delighted the authors present a relatively recent, improved, shorter version.

Although I believe this book will be most appreciated at the second and higher year graduate level, it should be easily understood by undergraduates who have taken a basic introductory complex variables course. The use of complex variables as a proof method would have value in several upper level undergraduate courses. The book has no exercises but a good instructor may use the theme of the book — complex variable approaches to proofs — to enrich any course. Furthermore, it is traditional in many courses, including undergraduate courses, to have relevant bibliographies on reserve in the (math) library for those students who wish to spend more time learning the material. *Complex Proofs of Real Theorems *would be a welcome addition to any such reserve library collection.

Russell Jay Hendel ([email protected]) holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.