[Math] Functions of one complex variable: geometric theory

ag.algebraic-geometrybookscv.complex-variablestextbook-recommendation

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

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.

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