Topics in Complex Function Theory, Volume 3

Develops the higher parts of function theory in a unified presentation. Starts with elliptic integrals and functions and uniformization theory, continues with automorphic functions and the theory of abelian integrals and ends with the theory of abelian functions and modular functions in several variables. The last topic originates with the author and appears here for the first time in book form. ABELIAN FUNCTIONS.

Power Series in Several Variables.

The Preparation Theorem.

Regular Functions.

Meromorphic Functions.

The Theorem of Weierstrass and Cousin.

The Period Group.

Jacobian Functions.

Linearization of the Exponent System.

The Period Relations.

The Reduced Exponent System.

Existence Proofs.

Picard Varieties.

The Addition Theorem.

MODULAR FUNCTIONS OF SEVERAL VARIABLES.

Automorphic Functions of Several Variables.

Algebraic Relations Between Automorphic Functions.

Symplectic Geometry.

Abelian Functions and Modular Functions.

The Fundamental Region of the Modular Group.

Modular Forms.

The Field of Modular Functions.

Algebraic Dependence.

Bibliography.

Cumulative Index Volumes I, II, and III.

Carl Ludwig Siegel was born on December 31, 1896 in Berlin. He studied mathematics and astronomy in Berlin and Gttingen and held chairs at the Universities of Frankfurt and Gttingen before moving to the Institute for Advanced Study in Princeton in 1940. He returned to Gttingen in 1951 and died there in 1981.
Siegel was one of the leading mathematicians of the twentieth century, whose work, noted for its depth as well as breadth, ranged over many different fields such as number theory from the analytic, algebraic and geometrical points of view, automorphic functions of several complex variables, symplectic geometry, celestial mechanics.