Basic Complex Analysis

The book reveals complex analysis as a very elegant and lovely branch of mathematics. The level of rigour is not that of Marsden's other book, Elementary Classical Analysis. Instead, Basic Complex Analysis can be usefully read by non-maths majors, especially those in physics and engineering. Key ideas are well covered. Starting with the Laurant series, which generalises the Taylor series. Then, from this, the idea of contour integration is examined. Giving rise to the Residue Theorem and the winding number. All because the only term that does not integrate to 0 is 1/z, which gives the complex log and its imaginary argument is the only thing left. So simple and powerful. Amazing that an essentially arbitrarily intricate contour integral can be given by the residues at the enclosed poles! Yet the text's derivation should get straightforward to follow for most readers. If you are going onto advanced physics, like quantum electrodynamics, then this theorem is used extensively. The book also covers important subsequent ideas. Especially conformal mapping and the Schwartz-Christoffel transformation. The treatment of conformal mapping, though, is only a hint of the richness of analysis available here. The numerous problems are also good for the student to tackle.

When I first started with this book, I was not a fan. However, the book grew on me over time. Marsden and Hoffman do a very good job of blending both theoretical and computational aspects of complex analysis. They do a very good job of motivating and explaining the proofs, and they do not leave out any details (this is both good and bad - it can distracting, but as long as you pay attention, you will never get lost). The illustrations in the text are for the most part illuminating and useful, and the worked examples at the end of each section are not bad as well. I did have a few minor problems, though. While many of the exercises are good, some of them seemed rather trivial. The chapter on conformal mapping could use some work. The binding on mine started to come apart by the end of the semester, although that may have been my fault.

I used an earlier edition of this text as an instructor 20 years ago. The students in my class at the time were equally divided among the fields of mathematics, physics, and engineering. The book proved to be quite useful for all of them. Marsden skillfully strikes a balance between the needs of math majors preparing for graduate study and the needs of physics and engineering students seeking applications of complex analysis.The book is clearly written and well-organized, with plenty of examples and exercises. My only significant criticism of the first edition was the author's tendency to label many examples of contour integration as theorems. Technically, there is nothing wrong this, but I found that some of my students tended to memorize the statements of these "theorems" rather than focus on the methods of integration discussed (for example, "Pac-Man" integrals with branch cuts along rays other than the positive real axis). Nonetheless, this is a fine text that has--not surprisingly--continued to be widely used for over two decades.

I learned Complex Analysis from the 2nd ed. of the book and am definitely going to teach my class next year from the 3rd ed. It is easy to read and has examples and illustrations not found in Alfors, Rudin, or Cartan. Despite the other reader's comments, I believe that it is a wonderful book for students who are bright and for those who need help in comprehending new subjects.

It is totally beyond me why this textbook on this rather difficult subject is not prevalent everywhere. The best start to the enticing area of complex analysis in existence. Prepares you perfectly for more complex texts. Fantastic.