Hadamard gap theorem

The Hadamard gap theorem gives conditions under which a power series with sufficiently rapidly increasing exponents has its circle of convergence as a natural boundary.

Power series: An expression of the form ∑ aₙzⁿ, used to represent functions through coefficients and powers of a variable. The theorem concerns analytic functions represented by power series with selected nonzero coefficients.
Complex analysis: The mathematical study of functions of complex variables and their analytic properties. The theorem is a result about analytic functions and their domains in the complex plane.
Jacques Hadamard: A French mathematician whose work spanned analysis, geometry, and mathematical physics. The gap theorem bears his name and belongs to the theory of analytic functions he developed.
Weierstrass function: A continuous function defined by a trigonometric series that can be nowhere differentiable for suitable parameters. Its classical constructions illustrate how lacunary series can exhibit extreme boundary behavior.
Radius of convergence: The radius of the largest disk centered at a series' expansion point in which the series converges. The theorem identifies this radius's boundary as a barrier to analytic continuation.
Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Power series define holomorphic functions inside their disks of convergence.
Georg Fabry: A French mathematician known for work on power series and the distribution of their singularities. Fabry's gap theorem is a closely related result on natural boundaries of power series.
Lacunary Fourier series: A Fourier series whose nonzero frequencies are separated by large gaps. The theorem's sparsity principle also informs boundary questions for separated frequency series.
Natural boundary: A boundary across which a function cannot be analytically continued through any point. This is the conclusion imposed on the circle by the theorem's gap condition.
Taylor series: A power series whose coefficients are determined by derivatives of a function at an expansion point. The theorem concerns the boundary behavior of functions represented by power series.

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