zfermim12.f

ZFERMIM12 evaluates the complete Fermi–Dirac integral of order $-\tfrac12$, $$F_{-1/2}(x) = \int_0^\infty \frac{t^{-1/2}}{e^{t - x} + 1}\,dt,$$ by the rational-function approximations of Antia (1993, ApJS 84, 101); its comment gives a maximum error of $1.23\times10^{-12}$.

Called by SNEUT (neutrino losses, by its name). No COMMON blocks.

For $x < 2$, with $z = e^x$, $$F_{-1/2}(x) = z\,\frac{P_7(z)}{Q_7(z)},$$ and for $x \ge 2$, with $z = 1/x^2$, $$F_{-1/2}(x) = \sqrt{x}\,\frac{P_{11}(z)}{Q_{11}(z)},$$ where the polynomials' coefficients are a1, b1 (degree 7) and a2, b2 (degree 11). Both numerators have leading coefficient 1: the loops start from xx + a1(m1) and xx + a2(m2), so a1(8) and a2(12), both 1, are not read.

The convention. The integral is Antia's, without the factor $1/\Gamma(k+1)$: the coefficients give $F_{-1/2} \to \Gamma(\tfrac12)\,e^x$ for small $x$ and $2\sqrt x$ for large $x$. SNEUT uses it in the same convention: there $d\nu/dF_{1/2} = 1/\bigl(\tfrac12F_{-1/2}(\nu)\bigr)$, and the argument it passes to IFERMI12, $1.1052\times10^8\,\rho Y_e/T^{3/2}$, equals $n_e h^3/[4\pi(2m_e kT)^{3/2}]$ in it.