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absolutely convergent according to Theorem addition-theorem algebraic function analytic function arbitrarily arbitrary entire function axis of reals bear the value bearers boundary branch-points circle of convergence completely complex number Consequently constant construct continuation corresponding critical points definite denote domain of values doubly periodic functions elliptic function entire plane entire rational function equation Example expansion factor finite limit point finite number Fm(z following theorem func function f(z functional element functional values fundamental parallelogram half-plane hence infinite product integral lattice points meromorphic function Mittag-Leffler's Mittag-Leffler's theorem multiple-valued function negative powers number of negative obtained pair of primitive period-parallelogram period-points point z0 position and order power series precisely prescribed primitive period Proof prove regular function Riemann sphere Riemann surface roots sheet simply periodic functions single-valued and regular single-valued function singular points tion unit circle value of log Weierstrass's factor-theorem z-plane