Module reduced 

Reduced integrals F_n(v, t) of the reciprocal terms.

Upstream: _redincgamma and _redintkambe of lattice/_esum.pyx. The series in t, the continuation to other sheets and the rounding bounds are treams-rs extensions.

Structs

Reduced 🔒

Reduced integrals F_n(v, t) of one reciprocal point, as jets.

Constants

HALF_INTEGER_SERIES_T 🔒

Largest |t| at which Reduced sums the half-integer orders from their series in t (which loses up to e^|t| ulps) instead of the Kambe chain (whose correlated errors the sums cancel less the smaller |t|).

SERIES_REACH 🔒

Largest |t| at which Reduced takes the series for the integer orders below the base pair of the Kambe recurrence (2n < -2), which need a series of gamma functions of their own each, while the series in t takes all orders from one ladder.

SERIES_T 🔒

Largest |t| = |w|^2 at which Reduced sums every integer order of F_n(v, t) from its series in t: the Kambe integrals take eta = -i / w, whose powers leave the range of f64 from about |w| = 1e-50, and the series terms alternate at most like those of e^-|t|, losing a few ulps up to SERIES_REACH.

SERIES_TERMS 🔒

Terms of the series in t of Reduced at most; |t| <= SERIES_REACH stops within 20.

Functions

gamma_limit 🔒

reduced_gamma with the whole turns from the principal argument of v (on the lower side of the real axis) to its sheet, found once for many orders.

kambe_arguments 🔒

The argument x = sqrt(-2 v) w (Re x >= 0) and split eta = -i / w of the Kambe integrals with F_n(v, w^2) = 2 I_(2n-1)(x, eta) w^2n, for Re w >= 0.

on_sheet 🔒

F_n(v) of reduced_gamma from its value gamma on the principal branches, continued by turns (see gamma_limit) to its sheet.

principal_phase 🔒

z^n / (-z)^n on the principal branches, for z with a signed imaginary part: (-1)^n, or for half-integer n (-1)^floor(n) times -i on and below the real axis and i above it.

reduced_gamma 🔒

F_n(v) = Gamma(n, v) / (-v)^n (2n = twice_n), the w = 0 limit of reduced_kambe.

reduced_kambe 🔒

F_n(v, w^2) of integer n (the odd Kambe orders of 1D spherical sums) from Kambe integrals, with a first-order bound on its absolute rounding error in units of epsilon.

reduced_power 🔒

(-z)^n of reduced_gamma, with the principal root for half-integer n.

regular_bessel 🔒

x^-n J_n(x), entire in x^2, or NaN where AMOS fails.