5822b594

By: Kaapo Seppänen <seppanenkaapo@gmail.com>

compute h-function normalizations in native precision

The poly_exponential and poly_left_right_exponential normalizations were
binary64 constants, which limited the f128/Arb rescue path of Fermi-surface
localization to about 1e-16 relative accuracy. They are Bessel integrals:
sigma e^2 K_{(1-p)/2}(2) and 2 sigma e^2 K_{1-p}(2).
HFunctionSettings::normalization now evaluates them in the active precision.
Half-integer orders are an exact rational times sqrt(pi)/2. Integer orders
use the K_0(2) series, whose only transcendental input is Euler's constant,
the Wronskian for K_1 and the stable upward recurrence. h and h_dual share
this one definition; the supported powers are unchanged.

Tests compare the normalizations in f64, f128, 256-bit and Arb precision
against Symbolica's 1200-bit bessel_k, anchor the orders to the previous
tabulated values, and check that every profile integrates to one.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

Total

1s

Timeline