The gamma function of a real ball

arb_hypgeom_gamma, arb_hypgeom_rgamma and arb_hypgeom_lgamma compute $\Gamma(x)$, $1/\Gamma(x)$ and $\log\Gamma(x)$ for a real ball $x$; arb_hypgeom_gamma_fmpq and arb_hypgeom_gamma_fmpz take an exact rational or integer argument. They are the implementations behind arb_gamma, arb_rgamma, arb_lgamma, arb_gamma_fmpq and arb_gamma_fmpz. Each combines three methods: exact or hypergeometric formulas for rational arguments with small denominators, a tabulated Taylor series of $1/\Gamma$ at low and medium precision, and the Stirling series with argument reduction for large arguments and high precision. The algorithms are described with their error bounds in The gamma function and in [Joh2023g].

Values and domains#

• $\Gamma$ has poles at $0, -1, -2, \ldots$; if $x$ contains one, arb_hypgeom_gamma returns an indeterminate result.
• $1/\Gamma$ is entire; arb_hypgeom_rgamma returns exactly zero at the poles of $\Gamma$ and a finite enclosure for balls containing them (if necessary by computing $x \cdot (1/\Gamma(x+1))$).
• arb_hypgeom_lgamma is defined only for $x > 0$, where $\Gamma(x) > 0$; for other $x$ the result is indeterminate. The log-gamma function of negative and complex arguments, which is the principal branch $\log \Gamma$ (not $\log |\Gamma|$), is acb_hypgeom_lgamma.
• Ball exponents are arbitrary-precision integers, so the functions do not overflow: $\Gamma(10^{30})$ is a ball with a binary exponent of about $10^{32}$, and arb_hypgeom_lgamma computes $\log\Gamma(x)$ without forming $\Gamma(x)$ unless $x$ is an exact small fraction. Arguments larger than about $2^{10\,\mathit{prec}}$ give an indeterminate result.

Choice of method#

arb_hypgeom_gamma and arb_hypgeom_rgamma proceed as follows.

1. Exact small fractions. If $x$ is exact, $|x| < 2^{\mathit{prec}}$ and $4x$ is an integer (above 1000 bits also if $x$ has at most $\mathit{prec}/50$ fractional bits), $x$ is converted to a rational number and passed to arb_hypgeom_gamma_fmpq. Infinite and NaN midpoints, and the poles, are handled at this point too.
2. Taylor series. arb_hypgeom_gamma_taylor is tried next. It applies to $|x| < 160$, up to a working precision of 3456 bits, as long as the shift $r$ of the argument reduction (the nearest integer to $x$) satisfies $-(40 + (\mathit{prec}-40)/4) \le r \le 70 + (\mathit{prec}-40)/8$. In this range it is the fastest method by far: the work is one polynomial evaluation with tabulated coefficients and at most a short rising factorial.
3. Stirling series. Otherwise arb_hypgeom_gamma_stirling applies the reflection formula for $x < -5$, shifts the argument up to $x + r \approx \beta\,\mathit{prec}$ with $\beta \approx 0.2$, and sums the Stirling series.

For inexact $x$, the Taylor method evaluates at the midpoint and adds a bound for the propagated error, while the Stirling method lowers the working precision to the accuracy of the input plus 20 bits; very wide balls are bounded directly using the monotonicity of $\Gamma$ on each side of its minimum at $x \approx 1.4616$.

arb_hypgeom_gamma_fmpq uses the following methods for $x = p/q$:

• $\Gamma(n) = (n-1)!$ computed exactly for a positive integer $n$ when $n < 1200$ or $n \log n$ is small compared to the precision;
• for $q \in \{2, 3, 4, 6\}$ and $|p|$ not too large, reduction to $\Gamma(a)$ with $0 < a \le 1$ by a rising factorial, and $\Gamma(1/2) = \sqrt\pi$, $\Gamma(1/3)$, $\Gamma(1/4)$ and the reflection and multiplication formulas for the others; the two constants are computed once per precision and cached;
• for other denominators above about $7000 + 300 \log_2 q$ bits, a hypergeometric series for $\Gamma(a)$ summed by binary splitting, which costs $O(M(n) \log^2 n)$ at $n$ bits;
• otherwise the Taylor or Stirling series at the rational argument rounded to a ball.

arb_hypgeom_lgamma uses the same exact and Taylor methods followed by a logarithm, and otherwise a Stirling series for $\log\Gamma$, so that no exponential of a huge number is formed.

Example#

The program computes $\Gamma(1/3)$, $\Gamma$ of a ball, the values at a pole and $\log\Gamma(10^{30})$:

#include <flint/fmpq.h>
#include <flint/arb.h>
#include <flint/arb_hypgeom.h>

int main(void)
{
    arb_t x, y;
    fmpq_t q;

    arb_init(x);
    arb_init(y);
    fmpq_init(q);

    /* Gamma(1/3) from the cached constant */
    fmpq_set_si(q, 1, 3);
    arb_hypgeom_gamma_fmpq(y, q, 128);
    flint_printf("Gamma(1/3)      = "); arb_printn(y, 35, 0); flint_printf("\n");

    /* a ball argument: the result contains Gamma(t) for all t in x */
    arb_set_str(x, "3.25 +/- 1e-10", 128);
    arb_hypgeom_gamma(y, x, 128);
    flint_printf("Gamma(3.25+-e)  = "); arb_printn(y, 20, 0); flint_printf("\n");

    /* the reciprocal gamma function is zero at the poles */
    arb_set_si(x, -3);
    arb_hypgeom_rgamma(y, x, 128);
    flint_printf("1/Gamma(-3)     = "); arb_printn(y, 20, 0); flint_printf("\n");
    arb_hypgeom_gamma(y, x, 128);
    flint_printf("Gamma(-3)       = "); arb_printn(y, 20, 0); flint_printf("\n");

    /* log Gamma(10^30) does not overflow */
    arb_set_str(x, "1e30", 128);
    arb_hypgeom_lgamma(y, x, 128);
    flint_printf("lgamma(10^30)   = "); arb_printn(y, 38, 0); flint_printf("\n");

    arb_clear(x);
    arb_clear(y);
    fmpq_clear(q);
    flint_cleanup();
    return 0;
}

Gamma(1/3)      = [2.6789385347077476336556929409746776 +/- 4.42e-35]
Gamma(3.25+-e)  = [2.549256967 +/- 5.45e-10]
1/Gamma(-3)     = 0
Gamma(-3)       = nan
lgamma(10^30)   = [68077552789821370520539743640497.306390 +/- 3.10e-7]

For the ball $3.25 \pm 10^{-10}$, the radius of the result is of the order of $|\Gamma'(3.25)| \cdot 10^{-10}$: the error propagated from the input dominates, and the midpoint is printed only to the digits that the radius justifies. The value $\log\Gamma(10^{30}) \approx 6.8 \cdot 10^{31}$ has 126 correct bits, the accuracy requested.