gr_special.h

Set res to $\pi$, Euler's constant $\gamma$, Catalan's constant $G$, Khinchin's constant $K$ or Glaisher's constant $A$. Rings that do not contain the constant return GR_DOMAIN (gr_pi in $\mathbb{Q}$) or GR_UNABLE.

Set res to $e^x$, $e^x - 1$ (accurate for small $x$), $2^x$ or $10^x$.

Sets res to $e^{\pi i x}$. For rational $x$ this is a root of unity, computed exactly by qqbar.

Set res to the principal natural logarithm $\log x$, to $\log(1 + x)$ (accurate for small $x$), or to the logarithms in base 2 and 10. GR_DOMAIN for $x = 0$.

Sets res to $\log(x) / (\pi i)$, the inverse of gr_exp_pi_i on the principal branch.

Set res to the principal branch $W_0(x)$ of the Lambert W function, or to the branch $W_k(x)$ with index $k$ given by the fmpz_t argument [Joh2017b].

Set res to $\sin x$, $\cos x$, both at once (gr_sin_cos sets res1 to the sine and res2 to the cosine), $\tan x$, $\cot x$, $\sec x = 1/\cos x$ or $\csc x = 1/\sin x$. Poles give GR_DOMAIN.

The same functions evaluated at $\pi x$: $\sin(\pi x)$ and so on. They are exact at integers and half-integers in ball arithmetic, and algebraic for rational $x$, which qqbar and ca compute exactly.

Set res to $\operatorname{sinc}(x) = \sin(x)/x$, respectively $\sin(\pi x)/(\pi x)$, with the value 1 at $x = 0$.

Set res to the hyperbolic functions $\sinh x$, $\cosh x$ (both at once for gr_sinh_cosh), $\tanh x$, $\coth x$, $\operatorname{sech} x$ and $\operatorname{csch} x$.

Set res to $\operatorname{asin} x$, $\operatorname{acos} x$, $\operatorname{atan} x$, $\operatorname{acot} x = \operatorname{atan}(1/x)$, $\operatorname{asec} x = \operatorname{acos}(1/x)$ and $\operatorname{acsc} x = \operatorname{asin}(1/x)$.

Sets res to $\operatorname{atan2}(y, x)$, the argument of the point $(x, y)$, where the first argument is $y$ and the second $x$, as in C. Defined for real rings.

The inverse functions divided by $\pi$: $\operatorname{asin}(x)/\pi$ and so on, which are rational for many algebraic $x$.

Set res to the inverse hyperbolic functions.

Set res to the factorial $x! = \Gamma(x + 1)$. The vec version writes the first len factorials $0!, 1!, \ldots, (\mathit{len} - 1)!$ to the array res.

Set res to the reciprocal factorial $1/x! = 1/\Gamma(x+1)$, and the vec version to $1/0!, \ldots, 1/(\mathit{len} - 1)!$; GR_DOMAIN if a factorial is not invertible in the ring.

Set res to the binomial coefficient $\binom{x}{y}$. The vec versions write $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{\mathit{len}-1}$ to res. For a two-dimensional table (Pascal's triangle), gr_mat_pascal is faster than repeated calls.

Set res to the rising factorial $x (x+1) \cdots (x+y-1)$ and the falling factorial $x (x-1) \cdots (x-y+1)$, or their extensions $\Gamma(x+y)/\Gamma(x)$ and $\Gamma(x+1)/\Gamma(x-y+1)$ to non-integer $y$.

Set res to the gamma function $\Gamma(x)$, its reciprocal $1/\Gamma(x)$ (entire, zero at the poles of $\Gamma$), the log-gamma function $\log \Gamma(x)$ (the principal branch continuous away from the negative real axis) or the digamma function $\psi(x) = \Gamma'(x)/\Gamma(x)$. gr_gamma returns GR_DOMAIN at the poles $0, -1, -2, \ldots$. See The gamma function.

Sets res to the beta function $B(x, y) = \Gamma(x) \Gamma(y) / \Gamma(x + y)$.

Set res to the Barnes G-function $G(x)$, respectively its logarithm $\log G(x)$.

Set res to the double factorial $x!! = x (x - 2) (x - 4) \cdots$, ending at 1 or 2.

Set res to the harmonic number $H_x = \sum_{k=1}^{x} 1/k$, or $\psi(x+1) + \gamma$ for non-integer $x$.

Set res to the Bernoulli number $B_n$ (with $B_1 = -1/2$). See bernoulli and Bernoulli numbers.

Set res to the Euler number $E_n$ (the coefficients of $\operatorname{sech}(t) = \sum E_n t^n / n!$).

Set res to the Bernoulli polynomial $B_n(x)$ or the Euler polynomial $E_n(x)$. These methods are implemented for real and complex balls (arb_bernoulli_poly_ui, acb_bernoulli_poly_ui); other rings return GR_UNABLE.

Set res to the Fibonacci number $F_n$, with $F_0 = 0$ and $F_1 = 1$; gr_fib_fmpz accepts negative $n$, with $F_{-n} = (-1)^{n+1} F_n$.

Set res to the unsigned Stirling number of the first kind $\left[{x \atop y}\right]$, the signed Stirling number of the first kind $s(x, y)$, or the Stirling number of the second kind $\left\{{x \atop y}\right\}$ (Stirling and Bell numbers).

Write the row $S(x, 0), S(x, 1), \ldots, S(x, \mathit{len} - 1)$ of the respective Stirling numbers to res. For a two-dimensional table, gr_mat_stirling is faster than repeated calls.

Set res to the Bell number $B_n$, the number of partitions of a set of $n$ elements (Stirling and Bell numbers).

Set res to the number of partitions $p(n)$ of the integer $n$ (Computing the partition function).

Set res to the error function $\operatorname{erf}(x)$, the complementary error function $\operatorname{erfc}(x) = 1 - \operatorname{erf}(x)$, the scaled complementary error function $\operatorname{erfcx}(x) = e^{x^2} \operatorname{erfc}(x)$, or the imaginary error function $\operatorname{erfi}(x) = -i \operatorname{erf}(ix)$.

Set res to the inverse error function and the inverse complementary error function, for real arguments in $(-1, 1)$, respectively $(0, 2)$.

Set res to the Fresnel integral $S(x) = \int_0^x \sin(\pi t^2 / 2) \, dt$ or $C(x) = \int_0^x \cos(\pi t^2 / 2) \, dt$ if normalized is nonzero, and to the unnormalised integrals $\int_0^x \sin(t^2) \, dt$ and $\int_0^x \cos(t^2) \, dt$ otherwise; gr_fresnel sets res1 to $S$ and res2 to $C$.

Set res to the incomplete gamma functions with parameter $s = x$ and argument $z = y$: if regularized is 0, the upper function $\Gamma(s, z) = \int_z^\infty t^{s-1} e^{-t} \, dt$ and the lower function $\gamma(s, z) = \int_0^z t^{s-1} e^{-t} \, dt$; if 1, the regularized versions $Q(s, z) = \Gamma(s,z)/\Gamma(s)$ and $P(s,z) = \gamma(s,z)/\Gamma(s)$; if 2, $z^{-s} \Gamma(s, z)$ and $\gamma^*(s, z) = z^{-s} P(s, z)$, as for acb_hypgeom_gamma_upper and acb_hypgeom_gamma_lower.

Sets res to the incomplete beta function $B(a, b; z) = \int_0^z t^{a-1} (1-t)^{b-1} \, dt$ with $a = x$, $b = y$, or to the regularized version $I(a, b; z) = B(a, b; z)/B(a, b)$ if regularized is nonzero.

Set res to the generalised exponential integral $E_s(z) = \int_1^\infty e^{-zt} t^{-s} \, dt$ with $s = x$ and $z = y$, respectively the exponential integral $\operatorname{Ei}(x)$.

Set res to the sine integral $\operatorname{Si}(x)$, the cosine integral $\operatorname{Ci}(x)$, the hyperbolic sine integral $\operatorname{Shi}(x)$ or the hyperbolic cosine integral $\operatorname{Chi}(x)$.

Sets res to the logarithmic integral $\operatorname{li}(x) = \int_0^x dt / \log t$, or to the offset logarithmic integral $\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)$ if offset is nonzero.

Sets res to the dilogarithm $\operatorname{Li}_2(x)$.

Set res to the Bessel functions $J_\nu(z)$, $Y_\nu(z)$, $I_\nu(z)$ or $K_\nu(z)$ of order $\nu = x$ and argument $z = y$; gr_bessel_j_y sets res1 to $J_\nu(z)$ and res2 to $Y_\nu(z)$.

Set res to the scaled Bessel functions $e^{-z} I_\nu(z)$ and $e^{z} K_\nu(z)$, which do not overflow for large $z$.

Sets res to the first len Taylor coefficients of $J_\nu(z + t)$ at $t = 0$, that is, $J_\nu^{(k)}(z) / k!$ for $0 \le k < \mathit{len}$ (the jet of $J_\nu$ at $z$). This is a generic function, not a method: it computes $J_\nu(z)$ and $J_{\nu+1}(z)$ and then the higher coefficients by a three-term recurrence (https://fungrim.org/entry/9b2f38/), and uses the series expansion at $z = 0$ for integer $\nu$.

Set res to the Airy functions $\operatorname{Ai}(x)$, $\operatorname{Bi}(x)$ and their derivatives; gr_airy sets res1, res2, res3, res4 to $\operatorname{Ai}(x)$, $\operatorname{Ai}'(x)$, $\operatorname{Bi}(x)$, $\operatorname{Bi}'(x)$ (as acb_hypgeom_airy).

Set res to the $n$-th real zero $a_n$, $b_n$, $a_n'$ or $b_n'$ of $\operatorname{Ai}$, $\operatorname{Bi}$, $\operatorname{Ai}'$ or $\operatorname{Bi}'$, counted from $n = 1$ and ordered by decreasing value; GR_DOMAIN for $n < 1$.

Set res to the Coulomb wave functions $F_\ell(\eta, z)$, $G_\ell(\eta, z)$, $H^{+}_\ell(\eta, z)$ and $H^{-}_\ell(\eta, z)$ with $\ell = x$, $\eta = y$ and argument $z$; gr_coulomb computes all four (as acb_hypgeom_coulomb).

Set res to the Chebyshev polynomials $T_n(x)$ and $U_n(x)$, for an integer $n$ (the _fmpz versions, which work in any ring) or $n$ in the ring (as hypergeometric functions in ball arithmetic).

Set res to the Jacobi polynomial $P_n^{(a,b)}(z)$, the Gegenbauer polynomial $C_n^{(m)}(z)$, the associated Laguerre polynomial $L_n^{(m)}(z)$ or the Hermite polynomial $H_n(z)$, extended to non-integer $n$ as hypergeometric functions (as the corresponding acb_hypgeom functions).

Set res to the associated Legendre functions of the first and second kind $P_n^m(z)$ and $Q_n^m(z)$; type selects the branch cut convention: 0 for Ferrers' functions on $(-1, 1)$ and 1 for the functions on $\mathbb{C} \setminus (-\infty, 1]$ (types 2 and 3 of Mathematica), as in acb_hypgeom_legendre_p.

Sets res to the spherical harmonic $Y_n^m(\theta, \phi)$ for integers $n$ and $m$.

Sets root to the root of index $k$ of the Legendre polynomial $P_n(x)$, with $0 \le k < n$ and the roots ordered by decreasing value, and, if weight is not NULL, weight to the corresponding weight of Gauss–Legendre quadrature (as arb_hypgeom_legendre_p_ui_root). GR_DOMAIN if $k \ge n$.

Set res to the hypergeometric functions ${}_0F_1(a; z)$, ${}_1F_1(a; b; z)$ (Kummer's function $M(a, b, z)$) and ${}_2F_1(a, b; c; z)$, or to their regularized versions.

Sets res to the confluent hypergeometric function of the second kind $U(a, b, z)$ (Tricomi's function). flags is ignored.

Sets res to the generalised hypergeometric function ${}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z)$ with the parameters given by the vectors a and b, or its regularized version.

Set res to the Riemann zeta function $\zeta(s)$; GR_DOMAIN at the pole $s = 1$. The method of gr_zeta_ui is implemented for real and complex balls (arb_zeta_ui); other rings return GR_UNABLE, and gr_zeta with an integer argument can be used instead.

Sets res to the Hurwitz zeta function $\zeta(s, a) = \sum_{k \ge 0} (k + a)^{-s}$ with $s = x$, $a = y$ (Zeta functions, polylogarithms and L-functions).

Sets res to the polylogarithm $\operatorname{Li}_s(z)$ with $s = x$, $z = y$.

Sets res to the polygamma function $\psi^{(s)}(z)$ with $s = x$, $z = y$, extended to complex $s$ as in acb_polygamma.

Sets res to the Lerch transcendent $\Phi(z, s, a) = \sum_{k \ge 0} z^k (k + a)^{-s}$ with $z = x$, $s = y$, $a = z$.

Sets res to the generalised Stieltjes constant $\gamma_n(a)$, the coefficients of the Laurent expansion of $\zeta(s, a)$ at $s = 1$.

Set res to the Dirichlet eta function $\eta(s) = (1 - 2^{1-s}) \zeta(s)$ and the Dirichlet beta function $\beta(s) = \sum_{k \ge 0} (-1)^k (2k+1)^{-s}$. gr_dirichlet_beta is implemented for real and complex balls, as the Dirichlet L-function of the nontrivial character modulo 4 (acb_dirichlet_l).

Sets res to the Riemann xi function $\xi(s) = \tfrac{1}{2} s (s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)$.

Set res to the $n$-th nontrivial zero $\rho_n = 1/2 + i t_n$ of $\zeta(s)$ in the upper half-plane, counted from $n = 1$ by increasing height; to the len consecutive zeros $\rho_n, \rho_{n+1}, \ldots$; or to the number $N(t)$ of zeros with imaginary part in $(0, t]$. The ball implementations return GR_UNABLE for $n < 1$ and for non-real or non-finite $t$.

Set res to the value $\chi(n)$ of the Dirichlet character chi of the group G, respectively the values $\chi(0), \ldots, \chi(\mathit{len}-1)$, as complex numbers. These are not methods: they are implemented for complex balls (acb) only and return GR_UNABLE otherwise. Declared only if dirichlet.h is included first.

Sets res to the Dirichlet L-function $L(s, \chi)$ for the character chi of the group G, in complex balls, in real balls (GR_UNABLE unless the value is real), or as a power series in gr_series rings (gr_series_dirichlet_l); GR_UNABLE in other rings.

Set res to the Hardy theta function $\theta(t)$ and Hardy Z-function $Z(t)$ of the character chi, defined so that $Z(t) = e^{i \theta(t)} L(1/2 + it, \chi)$ is real for real $t$, with the same rings as gr_dirichlet_l.

Set res to the Jacobi theta functions $\theta_j(z, \tau)$, with the argument scaled by $\pi$ as in acb_modular_theta (for example $\theta_3(z, \tau) = \sum_{n \in \mathbb{Z}} q^{n^2} e^{2\pi i n z}$); gr_jacobi_theta computes all four.

Set res to the modular $j$-invariant $j(\tau)$, the modular lambda function $\lambda(\tau)$ or the modular discriminant $\Delta(\tau)$.

Set res to the Dedekind eta function $\eta(\tau)$. gr_dedekind_eta_q takes the nome $q = e^{2 \pi i \tau}$ instead and computes $\eta$ without the leading factor $q^{1/24}$, that is, $q^{-1/24} \eta(\tau) = \prod_{k \ge 1} (1 - q^k)$ (as acb_modular_eta_sum does), which is a single-valued function of $q$ for $|q| < 1$. It returns GR_DOMAIN if $|q| \ge 1$, and is implemented for real and complex balls.

Sets res to the value $H_D(x)$ of the Hilbert class polynomial for the discriminant $D$ at x. If ctx is a gr_poly ring and x its generator, the polynomial itself is returned. GR_DOMAIN if $D$ is not a negative discriminant. The default computes $H_D$ with acb_modular_hilbert_class_poly and evaluates it, so it works in any ring.

Set res to the Eisenstein series $G_k(\tau) = \sum_{(m,n) \ne (0,0)} (m + n\tau)^{-k}$ or its normalisation $E_k(\tau) = G_k(\tau) / (2 \zeta(k))$, for even $k \ge 2$ (GR_DOMAIN for odd $k$ or $k = 0$); gr_eisenstein_g_vec writes $G_4, G_6, \ldots, G_{2\mathit{len}+2}$.

Set res to the arithmetic-geometric mean $M(x, y)$, respectively $M(1, x)$.

Set res to the complete elliptic integrals of the first, second and third kind $K(m)$, $E(m)$ and $\Pi(n, m)$.