# functions of one complex variable: geometric theory

Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English?

When I studied complex analysis, I used two textbooks:

1. An excellent texbook by A. Hurwitz (with collaboration of R. Courant), "Vorlesungen über allgemeine funktionentheorie und elliptische funktionen", ("Lectures on general theory of functions and elliptic functions"), in German, and

2. An excellent [though smaller] textbook by B. Shabat, "Introduction to complex analysis, vol. 1", in Russian, (Б.В. Шабат, "Введение в комплексный анализ").

Both books were never translated into English, besides a few chapters from the 2nd book, which were translated by Lenya Ryzhik.

For example, here are some topic in the geometric theory in a book by Hurwitz (and Courant):

• Riemann sphere; its automorphisms;
• conformal mappings;
• geometry of the maps $z^n$, $1/z$, $\exp(z)$ and $\log(z)$,
• algebraic functions given by $w^n = G(z)$, where $G(z)$ is a polynomial;
• Riemann surfaces of algebraic functions; examples thereof; Riemann-Hurwitz formula;
• analytic continuation;
• Schwarz' symmetry principle,
• Weierstrass' funcion $\wp(z,\tau)$ and the embedding of the complex torus $\mathbb{C}/L$ as a cubic curve into $\mathbb{P}^2$;
• the modular function $j(\tau)$.

What are the good English textbooks? Is there one textbook covering this, like one by Hurwitz?

Thank you

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There is no comparable book in English. (I mean Hurwitz-Courant) I am teaching the subject for about 30 years, and I was unable to find a similar book. But notice: the modern German version was heavily edited by Rohrl, and the Russian version by Evgrafov. Each of them very much improved the original which was full of mistakes. Someone has to do this for English readers:-) – Alexandre Eremenko Apr 3 '13 at 19:35
Thank you, Alexandre. What are good books on elementary complex analysis which are more or less geometric? – Maxim Leyenson Apr 3 '13 at 21:02
I remember looking at Siegel's "Topics in complex function theory" (several vols.) when I was a student, and it seemed very nice. Not sure if it fits all your criteria. – Donu Arapura Apr 5 '13 at 15:19

As I said in the remark there is no book comparable to the Russian edition of Hurwitz-Courant (Evgrafov was the editor of the Russian translation who improved the original very much).

There are 2 comprehensive Russian books covering much of geometric theory; both exist in English translation: Markushevich and Goluzin. Another book which covers a lot of geometric theory is Caratheodory (2 vols).

None of these has the theory of compact Riemann surfaces, but Shabat (which you like) also does not have it. I would say that Markushevich is a good replacement of Shabat. Goluzin can serve as a source of graduate courses.

Exposition of compact Riemann surfaces in Courant is unique, on my opinion.

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Wilhelm Schlag turned his notes from a first-year graduate course into a book of sorts. It covers a lot of the topics you listed and it has a bunch of good problems. A manuscript appears on his website here: http://www.math.uchicago.edu/~schlag/bookweb.pdf; the published version of the book can be found here: http://www.ams.org/bookstore-getitem/item=gsm-154

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John, thank you. The book looks very good. Will study. – Maxim Leyenson Apr 3 '13 at 21:08
Link is broken. – 5space Aug 7 '14 at 7:06

I suggest "Functions of One Complex Variable" I and II, by John B. Conway. It is especially well-written and cover a lot of topics, from elementary ones (volume I) to more advanced (volume II).

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Thank you, Malik! Will study it. – Maxim Leyenson Apr 3 '13 at 21:08

There are many wonderful books on complex analysis in English. In fact it is the one subject on which I used to say one could pick almost any book and be well served. Probably this is because it was well treated in the beginning and most books simply recast the classical theory.

My favorite is by Henri Cartan, superb and cheap. Another nice more elementary book is by Frederick Greenleaf. The book of Lang is also excellent.

As to geometry the book by Jones and Singerman: Complex functions, an algebraic and geometric viewpoint, is very well done. The classic book by Ford on Automorphic functions is also recommended.

I second the recommendation of the book by Rick Miranda, a book that is just a joy to read.

Finally, the original papers of Riemann are highly recommended.

Probably everything is in the two volumes of Einar Hille, the only basic reference I know to include the big Picard theorem.

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I said this before, but the best book ever (in any subject) is Ahlfors'. The chapter on the Weierstrass function (e.g.) is impeccable.

I have always claimed that the best way to study Complex Analysis correctly is to read Ahlfors for the theory, and Marsden for the worked examples and long list of computational exercises.

BTW, you will also find much interesting material in "Visual complex analysis" by T. Needham. It is full of geometric insights, although on the elementary side (and it takes loooong to get to the harder material).

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I am impressed. I like very much the section on infinite products in Ahlfors, but I think it is a very perceptive person who can detect much geometry in that work. – roy smith Apr 6 '13 at 2:12

A lot of this stuff is in Rick Miranda's book.

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Algebraic Curves and Riemann Surfaces – Steve Apr 3 '13 at 18:49
Steve, I just found a book by Rick Miranda called "Algebraic curves and Riemann surfaces". However, it assumes that elementary complex analysis is already known (definitely such things as the notion of holomorphic function and equivalence of several definitions of it), and geometry of maps like $z^n$) – Maxim Leyenson Apr 3 '13 at 18:50
-- i.e., it looks to be a nice book, but starts with more advanced things – Maxim Leyenson Apr 3 '13 at 18:51

Gamelin T.W. Complex analysis. Springer, 2001.

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I suggest mcmullen's lectures:complex analysis and advance complex analysis(in mcmullen's homepage),also there is a book:Hand book of complex analysis(geometric function theory)in 2 vols by R.KUHNAU,I think can be Useful for you.

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In addition to many excellent texts already mentioned, take a look at the following:

Remmert, Reinhold Classical topics in complex function theory. Translated from the German by Leslie Kay. Graduate Texts in Mathematics, 172. Springer-Verlag, New York, 1998. xx+349 pp. ISBN: 0-387-98221-3

It is divided into three parts: A. Infinite product and partial fraction series. B. Mapping theory. C. Selecta (including e.g the theorems of Bloch, Picard and Schottky, as well as theory of Runge, related to interpolation and approximation by polynomials).

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