SWhat changed
A power series may converge inside a circle while its function can still be continued beyond part of that circle. Fabry develops coefficient and sign-change criteria that can rule this out at every boundary point. His 1896 paper includes explicit quantitative sparsity conditions on the nonzero powers, with the finite positive convergence radius normalized to one. One such condition requires successive exponent gaps, divided by the logarithm of the exponent, to grow without bound. These results enlarge the known families of functions whose convergence circle is a natural boundary, building on Hadamard's earlier work.
Scientific scope
A power series may converge inside a circle while its function can still be continued beyond part of that circle. Fabry develops coefficient and sign-change criteria that can rule this out at every boundary point. His 1896 paper includes explicit quantitative sparsity conditions on the nonzero powers, with the finite positive convergence radius normalized to one. One such condition requires successive exponent gaps, divided by the logarithm of the exponent, to grow without bound. These results enlarge the known families of functions whose convergence circle is a natural boundary, building on Hadamard's earlier work. Infinitely many missing terms alone are not enough. The stated convergence-radius, coefficient and local sparsity assumptions matter; later formulations named Fabry's gap theorem are not all attributed to this original statement. The author's closing discussion of what happens for general arbitrary coefficients is explicitly exploratory, not a probability-one or category theorem. A natural boundary concerns analytic continuation, not necessarily divergence at every boundary point.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Infinitely many missing terms alone are not enough. The stated convergence-radius, coefficient and local sparsity assumptions matter; later formulations named Fabry's gap theorem are not all attributed to this original statement. The author's closing discussion of what happens for general arbitrary coefficients is explicitly exploratory, not a probability-one or category theorem. A natural boundary concerns analytic continuation, not necessarily divergence at every boundary point.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
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