Numerical Representation of the Incomplete Gamma Function of Complex Argument

TL;DR

The paper tackles the problem of efficiently evaluating the incomplete gamma function $F_m(z)$ for complex arguments with small integer indices $m$, a quantity central to electron repulsion integrals. It surveys a broad spectrum of numerical schemes, including hypergeometric-series representations, Laurent asymptotics, nonstandard power-series constructions, index interpolation, and generic integration approaches, then benchmarks their accuracy and speed. The key finding is that a Taylor-series expansion around a fixed grid in the complex $z$-plane, paired with a static matrix of higher derivatives, provides the most robust and efficient evaluation across the complex domain, outperforming other methods whose accuracy regions are more restricted or whose costs are higher. This approach enables fast, stable computations for moving-atom electronic structure problems and other applications requiring $F_m(z)$ with complex arguments, while the study also clarifies the complementary roles and limits of Laurent expansions and other representations in different regions of the $z$-plane.

Abstract

Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, Gautschi's approximation to the Faddeeva function, classical numerical methods of treating the standard integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in the z-plane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure often in use for real-valued z.

Figures (16)

• Figure 1: Contour levels of the number of valid decimal digits $d=4\ldots 18$ of $F_m(z)$ by the power series (\ref{['eq.Kumm']}) and (\ref{['eq.KummTay']}), if the power series is truncated after $n=30$ or $60$ terms.
• Figure 2: Contour levels of the number of valid decimal digits $d=2\ldots 12$ of $F_m(z)$, if the Laurent series (\ref{['eq.KummLau']}) is summed up to $n=|z|+a$.
• Figure 3: Contour levels of the number $n$ of terms needed to obtain an accuracy of $d=12$ or $17$ digits of $F_m(z)$, if the power series (\ref{['eq.KummTay']}) and the Laurent series (\ref{['eq.KummLau']}) are used complimentarily.
• Figure 4: Contour levels of the number of valid decimal digits $d=6\ldots 18$ of $F_m(z)$, if the Laurent series (\ref{['eq.KummLau']}) is summed up to $N=|z|+a$ and the convergent factor $\vartheta_N$ terminated as described in the text.
• Figure 5: Contour levels of the number of valid decimal digits $d=4\ldots 20$ of $F_m(z)$ by the power series (\ref{['eq.taylH']}), if the power series is truncated after $n=30$ or $40$ terms. For $m=0$ and $n=60$, $d$ is $\ge19.6$ in this $z$-domain.