Special Functions & Convergence Tests

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Abstractions about gamma- and zeta-related functions, fundamental analytic identities, numerical approximation, and root-based convergence criteria.

6 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.

• Bessel–Clifford function — Use an entire reciprocal-gamma-normalized power series whose derivative shift, differential equation, and square-root substitutions organize ordinary and modified Bessel functions.
• 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.
• Euler's identity — Relate the constants e, i, π, 1, and 0 through the exact equality e^(iπ)+1=0, obtained by evaluating Euler's complex-exponential formula at a half-turn in radians.
• 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.
• 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.
• 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.