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ANALYTIC CONTINUATION analytic functions arg P(t Binomial Series c e k complex lacunary Fourier continue g converges absolutely Cornell University define depending on q entire function entire plane equations fb(K finite number formula fractional Laurent series Green's formula half-plane of convergence highest homogeneous degree holomorphic functions infinite integer lacunary Fourier series lacunary power series lacunary series lckl Lemma Let P(t lm(w log q log+ lower bounds Math meromorphic functions Mi+1 nk+1 P(Rn p+l)R plane with possible polar coordinates polynomial positive number possible poles possible simple poles Proof Re(w Re(z real coefficients region region of convergence regular points satisfying sector sequence series development simply connected solution element Step sufficiently large Theorem 2.1 unit disk unit left vanishes variables Vp(K W.H.J. Fuchs Weiss wQ)J Zygmund