The Concept of a Riemann Surface

a₁ Abel's theorem Abelian Abelian differential admissible local coordinate algebraic analytic continuation analytic form analytic function arbitrary associated boundary c₁ closed curve closed differential closed path coefficients compact concept conformal maps constant contained continuously differentiable convergence coordinate system countable cover transformations defined Dirichlet integral Dirichlet principle divisor domain E₁ equation equivalent exists finite number fixed point follows function element Funktionen given grad harmonic function hence inequality integral function K₁ lattice linearly independent Math meromorphic functions multiples neighborhood obtain open set open unit disc parameter perfect covering surface plane points p₁ poles power series proof punched surface radius regular analytic regular elements representation Riemann surface simply connected singularities smooth sphere surface F t-plane t₁ theorem theory topological map two-dimensional manifold uniformizing variable v₁ vanishes Weierstrass y₁ z-sphere z₁ zero