Special Functions & Series Convergence

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Abstractions about analytic special functions and convergence tests, covering complex extensions of classical functions (the gamma function, Dirichlet eta function), computational approximation (Lanczos approximation), and series-convergence criteria (the ratio test, root test).

7 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Abel's Limit Theorem — Equating a convergent coefficient series with the radial boundary limit of its associated power series.
• Dirichlet Eta Function — Extend the alternating reciprocal-power series into an entire complex function tied to zeta by η(s)=(1−2^(1−s))ζ(s), gaining convergence on Re(s)>0 and a characteristic extra zero lattice.
• Gamma Function — The unique positive log-convex extension of shifted factorial on the positive reals, represented by Euler's integral and continued meromorphically to the complex plane with simple poles at the nonpositive integers and no zeros.
• General Dirichlet Series — A general Dirichlet series weights complex coefficients by exponential terms indexed by a freely chosen increasing frequency sequence.
• Lanczos Approximation — Evaluate the gamma function at fixed precision by factoring out its dominant asymptotic behavior and approximating the remaining analytic factor with a short precomputed rational sum.
• Ratio Test — Classify absolute convergence or divergence of a series from the eventual magnitude ratio of successive nonzero terms: below one converges, above one diverges, and equality to one is inconclusive.
• Root Test — Classify an infinite series using the limit superior of the nth roots of its term magnitudes: below one gives absolute convergence, above one gives divergence, and one is inconclusive.