Book:Hermann Weyl/The Concept of a Riemann Surface

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Hermann Weyl: The Concept of a Riemann Surface

Published $\text {1913}$, Dover Publications

ISBN 0-486-47004-0 (translated by Hermann Weyl)

Subject Matter


I. Concept and Topology of Riemann Surfaces
$\S 1$. Weierstrass' concept of an analytic function
$\S 2$. The concept of an analytic form
$\S 3$. The relation between the concepts "analytic function" and "analytic form"
$\S 4$. The concept of a two-dimensional manifold
$\S 5$. Examples of surfaces
$\S 6$. Specialization; in particular, differentiable and Riemann surfaces
$\S 7$. Orientation
$\S 8$. Covering surfaces
$\S 9$. Differentials and line integrals. Homology
$\S 10$. Densities and surface integrals. The residue theorem
$\S 11$. The intersection number

II. Functions on Riemann Surfaces
$\S 12$. The Dirichlet integral and harmonic differentials
$\S 13$. Scheme for the construction of the potential arising from a doublet source
$\S 14$. The proof
$\S 15$. The elementary differentials
$\S 16$. The symmetry laws
$\S 17$. The uniform functions on $\mathfrak F$ as a subspace of the additive and multiplicative functions on $\hat {\mathfrak F}$. The Riemann-Roch theorem
$\S 18$. Abel's theorem. The inversion problem
$\S 19$. The algebraic function field
$\S 20$. Uniformization
$\S 21$. Riemann surfaces and non-Euclidean groups of motions. Fundamental regions. Poincaré $\Theta$-series
$\S 22$. The conformal mapping of a Riemann surface onto itself