This book, immediately striking for its conciseness, is one of the most remarkable works ever produced on the subject of algebraic functions and their integrals. The distinguishing feature of the book is its third chapter, on rational functions, which gives an extremely brief and clear account of the theory of divisors....

A very readable account is given of the topology of Riemann surfaces and of the general properties of abelian integrals. Abel's theorem is presented, with some simple applications. The inversion problem is studied for the cases of genus zero and genus unity. The chapter on the reduction of singularities is very noteworthy.... A final chapter illustrates the general theory with some examples. In particular, constructive methods are given for treating algebraic relations which are of the third degree in one of the variables.... The arithmetic theory of algebraic functions is a good thing. In making its study easy, Bliss has performed a service which will win him the gratitude of an ever increasing number of readers.

--Bulletin of the American Mathematical Society

Readership

Graduate students and research mathematicians interested in algebraic functions and their integrals.

Table of Contents

• Single-valued analytic functions
• Algebraic functions and their expansions
• Rational functions
• The Riemann surface of an algebraic function
• Integrals of rational functions
• Abel's theorem
• Birational transformations
• The reduction of singularities by transformation
• Inversion of Abelian integrals
• Examples
• List of references