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A. F. Leont'ev algebraic number AMERICAN MATHEMATICAL SOCIETY analytic function arbitrarily small arbitrary asymptotic periods characteristic function circle common zeros constant coefficients converge uniformly define definition of asymptotic depend distinguishing functions u(x entire func entire function F(z equation L(F equations of infinite estimate finite type cr function of finite function of order functions Q(z,R given distinguishing functions growth of F(z Hence inequalities 60 infinite order infinite type InM(r integral sign Linear Differential Equations maximum modulus M(r modulus of F(z monotonely increasing operator L(F order of F(z particular solution period Tq periodic function periods of F(z positive constant proves our theorem r>rQ rate of growth rational real numbers representation right-hand side satisfy the following sequence solution Fq(z solution of L(F t/Tq is irrational taking moduli tends to infinity tion tire function uniform convergence zero of p(t zeros of 4>(t