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The Euler addition theorem
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abelian addition theorem analytic continuation analytic function arbitrary assertion basic periods boundary point branch points circle coefficients compact Riemann region complex conformal mapping constant continued analytically corresponding covering surface curve on 91 curves Cx Cx and C2 defined deformation denote Dirichlet integral domain double points edges elliptic function elliptic integral end point Figure finite follows function element fundamental group harmonic function homotopy class implies interior of 93 intersection inverse function issuing from 30 lemniscatic means meromorphic function minimizing sequence number sphere obtain p-function pair parameter path period parallelogram permutation plane point 30 points of 91 poles polygon polynomial power series proof prove punctured radius rational functions regular result Riemann region 91 Riemann surface 91 schlicht Section sheets simply connected single-valued subarc surface element un(z vanishes variable view of Theorem w-plane wk(z yields zeros