Ostrowski–Hadamard gap theorem

Ostrowski–Hadamard gap theorem In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence. The result is named after the mathematicians Alexander Ostrowski and Jacques Hadamard.

Contents 1 Statement of the theorem 2 See also 3 References 4 External links Statement of the theorem Let 0 < p1 < p2 < ... be a sequence of integers such that, for some λ > 1 and all j ∈ N, {displaystyle {frac {p_{j+1}}{p_{j}}}>lambda .} Let (αj)j∈N be a sequence of complex numbers such that the power series {displaystyle f(z)=sum _{jin mathbf {N} }alpha _{j}z^{p_{j}}} has radius of convergence 1. Then no point z with |z| = 1 is a regular point for f, i.e. f cannot be analytically extended from the open unit disc D to any larger open set including even a single point of the boundary of D.

See also Lacunary function Fabry gap theorem References Krantz, Steven G. (1999). Handbook of complex variables. Boston, MA: Birkhäuser Boston Inc. pp. 199-120. ISBN 0-8176-4011-8. MR1738432 External links Weisstein, Eric W. "Ostrowski–Hadamard gap theorem". MathWorld.

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