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:
-
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 -
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
Best Answer
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.