arb_hypgeom_gamma function

Gamma and Related Functions

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 [REMOVE_ME]$\int_{0}^{\infty} t^{z - 1} e^{-t} \text{d}tintegral_0^Inf t^(z - 1) exp(-t) dt [REMOVE_ME_2]$

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 [REMOVE_ME]$\int_{0}^{1} t^{a - 1} (1 - t)^{b - 1} \text{d}tintegral_0^1 t^(a - 1) (1 - t)^(b - 1) dt [REMOVE_ME_2]$

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

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

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

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

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.

Returns

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.

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.

See Also

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

References

The FLINT documentation of the underlying

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

Examples

## TODO