Cover; Title Page; Copyright Page; Dedication; Preface; Contents; I. Concept and Topology of Riemann Surfaces; 1. Weierstrass' concept of an analytic function; 2. The concept of an analytic form; 3. The relation between the concepts "analytic function" and "analytic form"; 4. The concept of a two-dimensional manifold; 5. Examples of surfaces; 6. Specialization; in particular, differentiable and Riemann surfaces.; 7. Orientation; 8. Covering surfaces; 9. Differentials and line integrals. Homology.;This classic on the general history of functions was written by one of the twentieth century's best-known mathematicians. Hermann Weyl, who worked with Einstein at Princeton, combined function theory and geometry in this high-level landmark work, forming a new branch of mathematics and the basis of the modern approach to analysis, geometry, and topology. The author intended this book not only to develop the basic ideas of Riemann's theory of algebraic functions and their integrals but also to examine the related ideas and theorems with an unprecedented degree of rigor. Weyl's two-part treatment begins by defining the concept and topology of Riemann surfaces and concludes with an exploration of functions of Riemann surfaces. His teachings illustrate the role of Riemann surfaces as not only devices for visualizing the values of analytic functions but also as indispensable components of the theory.