SWhat changed
Approximating a complex analytic function is useful only if its important features remain controlled. Hurwitz proves that when holomorphic functions converge uniformly near every interior point to a limit that is not identically zero, their zeros track the zeros of the limit locally. Around an isolated zero of multiplicity r, choose a small boundary containing no limit zero. All sufficiently late approximants then have exactly r zeros inside, counted with multiplicity. The original paper develops this general tool before using it for Bessel functions and explicitly includes polynomial partial sums of a power series.
Scientific scope
Approximating a complex analytic function is useful only if its important features remain controlled. Hurwitz proves that when holomorphic functions converge uniformly near every interior point to a limit that is not identically zero, their zeros track the zeros of the limit locally. Around an isolated zero of multiplicity r, choose a small boundary containing no limit zero. All sufficiently late approximants then have exactly r zeros inside, counted with multiplicity. The original paper develops this general tool before using it for Bessel functions and explicitly includes polynomial partial sums of a power series. Pointwise convergence is not enough. The limit must not be identically zero, and the chosen local boundary must avoid its zeros. The theorem does not assign permanent labels to individual roots or control roots escaping to the domain boundary or infinity. This record covers the finite holomorphic local result; pole remarks and detailed Bessel zero-counting conclusions are outside its scope. The contour-comparison input and earlier Bessel work retain separate credit.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Pointwise convergence is not enough. The limit must not be identically zero, and the chosen local boundary must avoid its zeros. The theorem does not assign permanent labels to individual roots or control roots escaping to the domain boundary or infinity. This record covers the finite holomorphic local result; pole remarks and detailed Bessel zero-counting conclusions are outside its scope. The contour-comparison input and earlier Bessel work retain separate credit.
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.
Provisional WRS assessment, edition 2026-10-23.7. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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