arb_hypgeom_gamma: Gamma and Related Functions

In flint: Fast Library for Number Theory

Gamma and Related Functions

Description

Compute the gamma function, the reciprocal gamma function, the logarithm of the absolute value of the gamma function, the polygamma function, or the beta function. The gamma function \Gamma(z) is defined by

\int_{0}^{\infty} t^{z - 1} e^{-t} \text{d}t

for \Re(z) > 0 and by analytic continuation elsewhere in the z-plane, excluding poles at z = 0, -1, \ldots. The beta function B(a, b) is defined by

\int_{0}^{1} t^{a - 1} (1 - t)^{b - 1} \text{d}t

for \Re(a), \Re(b) > 0 and by analytic continuation to all other (a, b).

Usage

arb_hypgeom_gamma(x, prec = flintPrec())
acb_hypgeom_gamma(z, prec = flintPrec())

arb_hypgeom_rgamma(x, prec = flintPrec())
acb_hypgeom_rgamma(z, prec = flintPrec())

arb_hypgeom_lgamma(x, prec = flintPrec())
acb_hypgeom_lgamma(z, prec = flintPrec())

## arb_hypgeom_polygamma(s = 0, z, prec = flintPrec())
   acb_hypgeom_polygamma(s = 0, z, prec = flintPrec())

arb_hypgeom_beta(a, b, prec = flintPrec())
acb_hypgeom_beta(a, b, prec = flintPrec())

Arguments

x, z, s, a, b	numeric, complex, arb, or acb vectors.
prec	a numeric or slong vector indicating the desired precision as a number of bits.

Details

acb_hypgeom_polygamma(s, z) evaluates the polygamma function of order s at z. The order s can be any complex number. For nonnegative integers m, s = m corresponds to the derivative of order m of the digamma function \psi(z) = \Gamma'(z)/\Gamma(z). Use acb_hypgeom_polygamma(0, z) to evaluate the digamma function at z.

Value

An arb or acb vector storing function values with error bounds. Its length is the maximum of the lengths of the arguments or zero (zero if any argument has length zero). The arguments are recycled as necessary.

References

The FLINT documentation of the underlying C functions: https://flintlib.org/doc/arb_hypgeom.html, https://flintlib.org/doc/acb_hypgeom.html

NIST Digital Library of Mathematical Functions: https://dlmf.nist.gov/5

See Also

Classes arb and acb; arb_hypgeom_gamma_lower and arb_hypgeom_beta_lower for the “incomplete” gamma and beta functions.

Examples

## TODO

flint documentation built on June 8, 2025, 1:27 p.m.