Function reduced_gamma 

pub(super) fn reduced_gamma(
    twice_n: i32,
    value: Complex,
    sheet: Option<Sheet>,
) -> Complex

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

Branch

On the principal branches, or continued to the argument sheet of v, which callers pass for Im k >= 0: there the outgoing sums take the phase v^n / (-v)^n = e^(-i pi n) of the lower side of the real axis.

Formula

A turn delta of the sheet adds -delta v^-n / (-n)! to integer n <= 0. Half-integer orders change sign above the real axis and lose 2 Gamma(n) / (-v)^n after an odd number of turns (DLMF 8.2.10). That term solves the homogeneous degree recurrence, so the continued F_n keep their derivative identities and recurrences.