arb_hypgeom.h

Sets res to the rising factorial $(x)_n = x (x+1) \cdots (x+n-1)$, with $(x)_0 = 1$; arb_hypgeom_rising allows any real $n$ and computes $\Gamma(x+n)/\Gamma(x)$.

Sets the vector res of length len to the coefficients of the polynomial $(x + t)_n = (x+t)(x+t+1)\cdots(x+t+n-1) \in \mathbb{R}[t]$, truncated to length len if $\mathit{len} < n+1$ and padded with zeros if $\mathit{len} > n+1$. The coefficient of $t^k$ is the $k$-th derivative of $(x)_n$ with respect to $x$ divided by $k!$.

Sets res to the gamma function $\Gamma(x)$. The result is indeterminate if $x$ contains a pole $0, -1, -2, \ldots$.

Sets res to the reciprocal gamma function $1/\Gamma(x)$, which is entire and zero at the nonpositive integers.

Sets res to $\log \Gamma(x)$ for $x > 0$. The result is indeterminate unless $x$ is positive; acb_hypgeom_lgamma computes the log-gamma function of negative and complex arguments.

Sets res to the sum $\sum_{n=1}^{N-1} B_{2n} / (2n(2n-1) z^{2n-1})$ of the Stirling series for $\log \Gamma(z)$, truncated before the term of index $N$. No bound for the truncation error is added.

Sets res to $\Gamma(x)$, or $1/\Gamma(x)$ if reciprocal is nonzero, computed with the Stirling series after argument reduction.

Attempts to compute $\Gamma(x)$, or $1/\Gamma(x)$ if reciprocal is nonzero, using a precomputed Taylor series. Returns 1 and sets res on success; returns 0 without changing res if $x$ or prec is too large for the method.

Sets res to the central binomial coefficient $\binom{2n}{n}$.

Sets res to the generalized hypergeometric function ${}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z)$, or the regularized function ${}_pF_q / (\Gamma(b_1) \cdots \Gamma(b_q))$ if regularized is nonzero. Wrapper of acb_hypgeom_pfq.

Sets res to the confluent hypergeometric limit function ${}_0F_1(a, z)$, or ${}_0F_1(a,z)/\Gamma(a)$ if regularized is nonzero. Wrapper of acb_hypgeom_0f1.

Sets res to Kummer's confluent hypergeometric function $M(a,b,z) = {}_1F_1(a,b,z)$, or $\mathbf{M}(a,b,z) = {}_1F_1(a,b,z)/\Gamma(b)$ if regularized is nonzero. The two names denote the same function, a wrapper of acb_hypgeom_m.

Sets res to Tricomi's confluent hypergeometric function $U(a,b,z)$, which is real for $z > 0$. Wrapper of acb_hypgeom_u.

Computes ${}_1F_1(a,b,z)$ (regularized if regularized is nonzero) by rigorous numerical integration of

The result is finite only if $a \ge 1$ and $b - a \ge 1$.

Computes $U(a,b,z)$ by rigorous numerical integration of

The result is finite only if $a \ge 1$ and $z > 0$. acb_hypgeom_u uses it as a fallback for real arguments in the same way as arb_hypgeom_1f1_integration.

Sets res to the Gauss hypergeometric function ${}_2F_1(a,b,c,z)$, or $\mathbf{F}(a,b,c,z) = {}_2F_1(a,b,c,z)/\Gamma(c)$ if the bit ACB_HYPGEOM_2F1_REGULARIZED (= 1) is set in regularized. Wrapper of acb_hypgeom_2f1; the flags of that function, which declare that a parameter difference is an exact integer, can be set in regularized too. The value is real for $z < 1$; on the branch cut $z > 1$ the result is indeterminate unless the function is a polynomial.

Computes ${}_2F_1(a,b,c,z)$ (regularized if regularized is nonzero) by rigorous numerical integration of

The result is finite only if $b \ge 1$, $c - b \ge 1$ and $z < 1$, possibly after exchanging $a$ and $b$. acb_hypgeom_2f1 uses it as a fallback for real arguments.

Sets res to the error function $\operatorname{erf}(z) = \frac{2}{\sqrt\pi} \int_0^z e^{-t^2} dt$.

Sets res to the complementary error function $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$, accurately also for large positive $z$ where $\operatorname{erfc}(z)$ is tiny.

Sets res to the imaginary error function $\operatorname{erfi}(z) = -i \operatorname{erf}(iz) = \frac{2}{\sqrt\pi} \int_0^z e^{t^2} dt$. Wrapper of acb_hypgeom_erfi.

Sets res to the inverse error function $\operatorname{erf}^{-1}(x)$ for $-1 \le x \le 1$, or the inverse complementary error function $\operatorname{erfc}^{-1}(x) = \operatorname{erf}^{-1}(1-x)$ for $0 \le x \le 2$. The values at the endpoints are $\pm\infty$; arguments outside the domain give an indeterminate result.

Sets res to $\operatorname{erf}$, $\operatorname{erfc}$ or $\operatorname{erfi}$ of the power series $z$, truncated to length len.

Sets res1 to the Fresnel sine integral $S(z)$ and res2 to the Fresnel cosine integral $C(z)$; either output may be NULL. If normalized is 0, $S(z) = \int_0^z \sin(t^2) dt$ and $C(z) = \int_0^z \cos(t^2) dt$; if normalized is 1, $S(z) = \int_0^z \sin(\tfrac12 \pi t^2) dt$ and $C(z) = \int_0^z \cos(\tfrac12 \pi t^2) dt$, the convention of Abramowitz and Stegun and of the DLMF [NIST2012]. Wrapper of acb_hypgeom_fresnel.

Sets res1 and res2 to the Fresnel sine and cosine integrals of the power series $z$, truncated to length len. Either output may be NULL.

Sets res to the upper incomplete gamma function $\Gamma(s,z) = \int_z^\infty t^{s-1} e^{-t} dt$ if regularized is 0, to $Q(s,z) = \Gamma(s,z)/\Gamma(s)$ if regularized is 1, and to $z^{-s} \Gamma(s,z) = E_{1-s}(z)$ if regularized is 2. Wrapper of acb_hypgeom_gamma_upper.

Computes the upper incomplete gamma function (with the same meaning of regularized as arb_hypgeom_gamma_upper) from $\Gamma(s,z) = z^s e^{-z} U(1, 1+s, z)$, using arb_hypgeom_u_integration. The result is finite only for $z > 0$. acb_hypgeom_gamma_upper uses it as a fallback for real arguments.

Sets res to the upper incomplete gamma function $\Gamma(s, z)$ of the constant $s$ and the power series $z$, truncated to length $n$. regularized has the same meaning as in arb_hypgeom_gamma_upper.

Sets res to the lower incomplete gamma function $\gamma(s,z) = \int_0^z t^{s-1} e^{-t} dt = \frac{z^s}{s} {}_1F_1(s, s+1, -z)$ if regularized is 0, to $P(s,z) = \gamma(s,z)/\Gamma(s)$ if regularized is 1, and to $\gamma^{*}(s,z) = z^{-s} P(s,z)$, which is entire in both variables, if regularized is 2. Wrapper of acb_hypgeom_gamma_lower.

Sets res to the lower incomplete gamma function of the constant $s$ and the power series $z$, truncated to length $n$, with regularized as in arb_hypgeom_gamma_lower.

Sets res to the incomplete beta function $B(a,b;z) = \int_0^z t^{a-1} (1-t)^{b-1} dt$, or the regularized function $I(a,b;z) = B(a,b;z)/B(a,b;1)$ if regularized is nonzero. The precise definition by analytic continuation is given for acb_hypgeom_beta_lower, of which this is a wrapper.

Sets res to $B(a,b;z)$ (or $I(a,b;z)$ if regularized is nonzero) for constants $a, b$ and the power series $z$, truncated to length $n$.

Sets res to the generalized exponential integral $E_s(z) = z^{s-1} \Gamma(1-s, z)$. Wrapper of acb_hypgeom_expint.

Sets res to the exponential integral $\operatorname{Ei}(z) = -\int_{-z}^\infty e^{-t}/t \, dt$ (a principal value for $z > 0$), defined for real $z \ne 0$. Wrapper of acb_hypgeom_ei.

Sets res to the exponential integral of the power series $z$, truncated to length len.

Sets res to the sine integral $\operatorname{Si}(z) = \int_0^z \sin(t)/t \, dt$.

Sets res to the sine integral of the power series $z$, truncated to length len.

Sets res to the cosine integral $\operatorname{Ci}(z) = -\int_z^\infty \cos(t)/t \, dt$ for $z > 0$. The result is indeterminate unless $z$ is positive, since the value is complex for $z < 0$.

Sets res to the cosine integral of the power series $z$, truncated to length len.

Sets res to the hyperbolic sine integral $\operatorname{Shi}(z) = -i \operatorname{Si}(iz) = \int_0^z \sinh(t)/t \, dt$. Wrapper of acb_hypgeom_shi.

Sets res to the hyperbolic sine integral of the power series $z$, truncated to length len.

Sets res to the hyperbolic cosine integral $\operatorname{Chi}(z) = \gamma + \log z + \int_0^z (\cosh(t) - 1)/t \, dt$ for $z > 0$. The result is indeterminate unless $z$ is positive. Wrapper of acb_hypgeom_chi.

Sets res to the hyperbolic cosine integral of the power series $z$, truncated to length len.

Sets res to the logarithmic integral $\operatorname{li}(z) = \operatorname{Ei}(\log z)$ if offset is zero, or the offset logarithmic integral $\operatorname{Li}(z) = \operatorname{li}(z) - \operatorname{li}(2)$ if offset is nonzero, for $z \ge 0$. The result is indeterminate unless $z$ is nonnegative. Wrapper of acb_hypgeom_li.

Sets res to the logarithmic integral (the offset version if offset is nonzero) of the power series $z$, truncated to length len.

Sets res to the Bessel function of the first kind $J_\nu(z)$ or of the second kind $Y_\nu(z)$. Wrappers of acb_hypgeom_bessel_j and acb_hypgeom_bessel_y; the result is indeterminate where the value is not real, for example $Y_\nu(z)$ with $z < 0$.

Sets res1 to $J_\nu(z)$ and res2 to $Y_\nu(z)$, computed together, which is faster than two separate calls. Wrapper of acb_hypgeom_bessel_jy.

Sets res to the modified Bessel function of the first kind $I_\nu(z) = z^\nu (iz)^{-\nu} J_\nu(iz)$, or to the scaled function $e^{-z} I_\nu(z)$, which does not overflow for large $z$. Wrappers of acb_hypgeom_bessel_i and acb_hypgeom_bessel_i_scaled.

Sets res to the modified Bessel function of the second kind $K_\nu(z)$, or to the scaled function $e^z K_\nu(z)$, which does not underflow for large $z$. Wrappers of acb_hypgeom_bessel_k and acb_hypgeom_bessel_k_scaled.

Computes $I_\nu(z)$ or $K_\nu(z)$ (multiplied by $e^{-z}$ or $e^{z}$ if scaled is nonzero) by rigorous numerical integration of an integral representation. These functions are intended for large $\nu$, where the series expansions are slow; acb_hypgeom_bessel_i and acb_hypgeom_bessel_k use them as fallbacks for real arguments.

Sets res to $J_\nu(z)$ for the constant $\nu$ and the power series $z$, truncated to length len.

Sets ai, ai_prime, bi, bi_prime to the Airy functions $\operatorname{Ai}(z)$, $\operatorname{Ai}'(z)$, $\operatorname{Bi}(z)$, $\operatorname{Bi}'(z)$. Any of the outputs may be NULL, and omitting outputs speeds up the evaluation. Wrapper of acb_hypgeom_airy.

Sets the vectors ai and bi to the Taylor coefficients of $\operatorname{Ai}$ and $\operatorname{Bi}$ at the point $z$, truncated to length len. Either output may be NULL; $z$ must not alias the outputs.

Sets the outputs to the Airy functions and their derivatives evaluated at the power series $z$, truncated to length len. Any of the outputs may be NULL.

Sets a, a_prime, b, b_prime to the $n$-th real zero $a_n$, $a'_n$, $b_n$, $b'_n$ of $\operatorname{Ai}$, $\operatorname{Ai}'$, $\operatorname{Bi}$, $\operatorname{Bi}'$ respectively; any of them may be NULL. The zeros are negative and indexed by increasing magnitude starting with $n = 1$, as in the literature. An index $n \le 0$ raises an exception.

Sets F and G to the regular and irregular Coulomb wave functions $F_\ell(\eta, z)$ and $G_\ell(\eta, z)$. Either output may be NULL. An output is indeterminate if its complex value is not real. Wrapper of acb_hypgeom_coulomb, which gives the definitions.

Sets the vectors F and G to the Taylor coefficients of the Coulomb wave functions at the point $z$, truncated to length len. Either output may be NULL.

Sets F and G to the Coulomb wave functions evaluated at the power series $z$, truncated to length len. Either output may be NULL.

Sets res to the Chebyshev polynomial $T_\nu(z)$ or $U_\nu(z)$, the Jacobi polynomial $P_n^{(a,b)}(z)$, the Gegenbauer polynomial $C_n^{m}(z)$, the Laguerre polynomial $L_n^{m}(z)$ or the Hermite polynomial $H_\nu(z)$, or their extensions to non-integer degree. Wrappers of the functions of the same name in acb_hypgeom, which give the definitions.

Sets res to the associated Legendre function of the first kind $P_n^m(z)$ or of the second kind $Q_n^m(z)$, with the branch cut convention selected by type (0 or 1) as described for acb_hypgeom_legendre_p and acb_hypgeom_legendre_q. For $m = 0$ and a nonnegative integer $n < 2^{\mathtt{FLINT\_BITS} - 1}$, arb_hypgeom_legendre_p calls arb_hypgeom_legendre_p_ui; otherwise both functions wrap the complex versions.

Sets res to the Legendre polynomial $P_n(x)$ and res_prime to its derivative $P'_n(x)$. Either output may be NULL.

Sets res to the $k$-th root $x_k$ of the Legendre polynomial $P_n$, with the roots in decreasing order $1 > x_0 > x_1 > \cdots > x_{n-1} > -1$, and, if weight is not NULL, sets weight to the corresponding weight $w_k = 2 / ((1 - x_k^2) P'_n(x_k)^2)$ of Gauss–Legendre quadrature on $[-1, 1]$. Requires $0 \le k < n$. See Roots of Legendre polynomials and Gauss–Legendre nodes.

Sets res to the dilogarithm $\operatorname{Li}_2(z) = -\int_0^z \log(1-t)/t \, dt$ for $z \le 1$; the result is indeterminate for $z > 1$, where the value is complex. Wrapper of acb_hypgeom_dilog.

Sets res to the finite sum $\sum_{n=0}^{N-1} \frac{(a_1)_n \cdots (a_{\mathit{alen}})_n}{(b_1)_n \cdots (b_{\mathit{blen}})_n} w^n$ with rational parameters $a_i$ and $b_j$, where $w = z$, or $w = 1/z$ if reciprocal is nonzero. There is no implicit $n!$: include $b_j = 1$ for the usual ${}_pF_q$ normalisation.

Sets res1 and res2 to the real and imaginary parts of the finite sum $\sum_{n=0}^{N-1} \frac{(a_1)_n \cdots}{(b_1)_n \cdots} (iw)^n$ with $w = z$, or $w = 1/z$ if reciprocal is nonzero, using only real arithmetic.

Sets $c_0, \ldots, c_{n-1}$ to the coefficients of the monic polynomial $(X+k)_n = X^n + c_{n-1} X^{n-1} + \cdots + c_0$ (the leading 1 is not stored). Requires $n \ge 2$ and raises an exception otherwise.

For $1 \le n < N$, sets bound[n] to an exponent $e$ such that the $n$-th term $|B_{2n}/(2n(2n-1) z^{2n-1})|$ of the Stirling series is at most $2^e$, given an upper bound zinv for $|1/z|$. Does not check for overflow or underflow of the exponents.

Sets c to a shallow copy of the tabulated coefficient $c_i$ of $1/\Gamma(1+x)$, $1 \le i <$ ARB_HYPGEOM_GAMMA_TAB_NUM, truncated to prec bits (rounded up to whole limbs), and err (if not NULL) to a bound for the truncation error. Returns 0, without setting anything, if prec exceeds the precision of the table and 1 otherwise. c points into static data and must not be modified or cleared; $i$ is not checked.

Sets res to $\sum_{k=0}^{N-1} z^k / (a)_k$ with $a = p/q$, computed by rectangular splitting. Requires that $p + qN$ fits in a word. Used for the error function.

Sets res to $\sum_{k=0}^{N-1} (a)_k / z^k$ with $a = p/q$, computed by rectangular splitting. Requires that $p + qN$ fits in a word. Used for the asymptotic expansion of the error function.

Returns a number of terms $N$ of the asymptotic series of $\Gamma(a, z)$ at infinity that gives an absolute error at most abs_tol, and sets err to a bound for the truncation error. Returns $-1$ if the tolerance cannot be reached. Assumes $z > 0$.

Sets res to the asymptotic series of $\Gamma(a,z)$ at infinity, including the prefactor $z^{a-1} e^{-z}$, truncated before the term of index $N$ and computed by binary splitting. The truncation error is not included.

Returns a number of terms $N$ of the Taylor series of $\gamma(a,z)$ at zero that gives an absolute error at most abs_tol, and sets err to a bound for the truncation error. Assumes $z > 0$.

Sets res to the series $\gamma(a,z) = \frac{z^a e^{-z}}{a} \sum_n \frac{z^n}{(a+1)_n}$ truncated before the term of index $N$ and computed by binary splitting. The truncation error is not included.

Returns a number of terms $N$ of the series at zero of $\Gamma(-n,z)$, for an integer $n \ge 0$, that gives an absolute error at most abs_tol, and sets err to a bound for the truncation error.

Sets res to the series at zero of $\Gamma(-n,z)$, for an integer $n \ge 0$, truncated before the term of index $N$ and computed by binary splitting. The truncation error is not included.

Sets res to the jet of $\Gamma(a + \varepsilon)$ for rational $a$ which is not a nonpositive integer, computed as $\gamma(a + \varepsilon, Z) + \Gamma(a + \varepsilon, Z)$ with an integer $Z \approx 0.55 \log(2) p$ at precision $p$, as in arb_hypgeom_gamma_fmpq: the lower series at $Z$ and the asymptotic series at $Z$ (at reduced precision, since $\Gamma(a+\varepsilon, Z)$ is small) are both evaluated by binary splitting. If the asymptotic series does not converge to the required accuracy (at low precision or for large $a$), $Z$ is increased until it does, or until the second term is negligible. With len = 1, this calls arb_gamma_fmpq.

Jet versions of _arb_hypgeom_gamma_upper_fmpq_inf_choose_N and _arb_hypgeom_gamma_upper_fmpq_inf_bsplit. The truncation error $(a-1)\cdots(a-N) \Gamma(a-N,z)$ is bounded using majorants for the polynomial factor and for $\int_z^{\infty} t^{a-N-1} e^{-t} \log^j(t) / j! \, dt$. The choose_N function requires $z \ge 1$ (otherwise it returns $-1$).

Jet versions of _arb_hypgeom_gamma_lower_fmpq_0_choose_N and _arb_hypgeom_gamma_lower_fmpq_0_bsplit.

Jet versions of _arb_hypgeom_gamma_upper_singular_si_choose_N and _arb_hypgeom_gamma_upper_singular_si_bsplit, computing the jet of $\Gamma(-n + \varepsilon, z)$ for an integer $n \ge 0$. This uses

If gamma1 is not NULL, it should contain the jet of $\Gamma(1 + \varepsilon)$ of length len + 1; otherwise this is computed internally.

Jet version of _arb_gamma_upper_fmpq_step_bsplit: given the jet Gz0 of $\Gamma(a + \varepsilon, z_0)$ and expmz0 containing $e^{-z_0}$, sets Gz1 to the jet of $\Gamma(a + \varepsilon, z_1)$. The matrix binary splitting is done with power series entries. The truncation error is bounded using Cauchy estimates on a circle of radius $R$ around $z_0$, with an extra factor $\max_{j < \text{len}} \Lambda^j / j!$ where $\Lambda$ bounds $|\log(t)|$ on the circle; the radius $R$ is chosen with this factor taken into account.

Given Gz0 and expmz0 containing $\Gamma(a, z_0)$ and $e^{-z_0}$, sets Gz1 to $\Gamma(a, z_1)$, computed from the Taylor series at $z_0$ by binary splitting with an absolute truncation error at most abs_tol (included in the output). Assumes $z_0, z_1 > 0$. This is one step of the bit-burst algorithm.

Attempts to compute $\operatorname{erf}(z)$, or $\operatorname{erfc}(z)$ if complementary is nonzero, with the bit-burst algorithm applied to $\Gamma(1/2, z^2)$. Returns 1 on success. Returns 0 without changing res if $z$ is not finite, if $|z| > \mathit{prec}$ or if $|z| < 2^{-\mathit{prec}/16}$.