Function asymptotic_series_g 

pub fn asymptotic_series_g<P, E, R, C, const SCALED: bool>(
    z: C,
    nu: R,
    far: C::Mask,
) -> C
where
    E: FloatElement,
    R: FloatVector<Element = E>,
    C: FloatVector<Mask = R::Mask> + TranscendentalMathWithPolicy + PrimalProjection<Primal = R> + BesselDetails<C> + Mul<R, Output = C>,
    P: Policy,

asymptotic_series_v over a general arithmetic: the argument z in C, the order in its real primal R, and the far mask supplied by the caller.

This is the body the real-order kernel runs, and through it the complex one. The loop is the same as asymptotic_series_v’s. It is a separate function rather than that one’s body because the integer and half-integer kernels call the real form from bounds that know nothing of BesselDetails, and the second-term machinery below is only meaningful off the real axis. Everything z-dependent is C arithmetic. The term numerators (2k+1)^2 - 4 nu^2 stay real and enter through C: Mul<R>, so a complex instantiation pays two real multiplies per term rather than a complex one.

The second exponential

The expansion has two exponential terms (DLMF 10.40.5), and the real line keeps one because the other is e^{-2x} relative, below epsilon anywhere this arm runs. That is a fact about the real axis. Off it the second term’s modulus is e^{-2 Re z}, which on the imaginary axis is one, so an arithmetic can ask for it through BesselDetails::ASYM_TWO_TERMS. It costs one more exponential and no extra series evaluation: the second sum is the first with every other sign flipped, so both are accumulated in the one loop.