arb_fpwrap.h

Compute $e^x$, using arb_exp and acb_exp.

Compute $e^x - 1$, accurately also for small $x$ (arb_expm1, acb_expm1).

Compute the natural logarithm (the principal branch for complex input), using arb_log and acb_log. The real version fails for $x \le 0$.

Compute $\log(1 + x)$, accurately also for small $x$ (arb_log1p, acb_log1p).

Compute $x^y$ (arb_pow, acb_pow); the complex version uses the principal branch $e^{y \log x}$.

Compute the square root (principal branch), using arb_sqrt and acb_sqrt.

Compute the reciprocal square root $x^{-1/2}$ (arb_rsqrt, acb_rsqrt).

Compute the cube root: the real cube root $x^{1/3}$ (arb_root_ui; this fails for negative $x$) and the principal complex cube root.

Compute $\sin x$ (arb_sin, acb_sin).

Compute $\cos x$ (arb_cos, acb_cos).

Compute $\tan x$ (arb_tan, acb_tan).

Compute $\cot x$ (arb_cot, acb_cot).

Compute $\sec x = 1/\cos x$ (arb_sec, acb_sec).

Compute $\csc x = 1/\sin x$ (arb_csc, acb_csc).

Compute $\operatorname{sinc} x = \sin(x)/x$ (arb_sinc, acb_sinc).

Compute $\sin(\pi x)$ (arb_sin_pi, acb_sin_pi), which is exact at integers.

Compute $\cos(\pi x)$ (arb_cos_pi, acb_cos_pi).

Compute $\tan(\pi x)$ (arb_tan_pi, acb_tan_pi).

Compute $\cot(\pi x)$ (arb_cot_pi, acb_cot_pi).

Compute $\operatorname{sinc}(\pi x) = \sin(\pi x)/(\pi x)$ (arb_sinc_pi, acb_sinc_pi).

Compute $\operatorname{asin} x$ (arb_asin, acb_asin).

Compute $\operatorname{acos} x$ (arb_acos, acb_acos).

Compute $\operatorname{atan} x$ (arb_atan, acb_atan).

Computes the two-argument arctangent $\operatorname{atan2}(x_1, x_2)$, the argument of the point $(x_2, x_1)$, as arb_atan2 with $y = x_1$ and $x = x_2$. It fails at the origin.

Compute $\operatorname{asinh} x$ (arb_asinh, acb_asinh).

Compute $\operatorname{acosh} x$ (arb_acosh, acb_acosh).

Compute $\operatorname{atanh} x$ (arb_atanh, acb_atanh).

Compute the Lambert W function on the given branch: for real input, branch must be 0 (the principal branch $W_0$) or $-1$ ($W_{-1}$); for complex input, any branch $k$ (arb_lambertw, acb_lambertw).

Compute the rising factorial $(x)_n = \Gamma(x + n)/\Gamma(x)$ (arb_rising, acb_rising).

Compute $\Gamma(x)$ (arb_gamma, acb_gamma).

Compute $1/\Gamma(x)$ (arb_rgamma, acb_rgamma), an entire function.

Compute $\log \Gamma(x)$ (arb_lgamma, acb_lgamma); the real version requires $x > 0$, the complex version gives the principal branch of the log-gamma function.

Compute the digamma function $\psi(x)$ (arb_digamma, acb_digamma).

Compute the Riemann zeta function $\zeta(x)$ (arb_zeta, acb_zeta).

Compute the Hurwitz zeta function $\zeta(s, z)$ (arb_hurwitz_zeta, acb_hurwitz_zeta).

Compute the Lerch transcendent $\Phi(z, s, a)$ (acb_dirichlet_lerch_phi); the real version fails if the value is not real.

Compute the Barnes G-function (acb_barnes_g; the real version fails if the result is not real).

Compute the logarithm of the Barnes G-function (acb_log_barnes_g).

Compute the polygamma function $\psi^{(s)}(z)$ (acb_polygamma).

Compute the polylogarithm $\operatorname{Li}_s(z)$ (arb_polylog, acb_polylog).

Computes the Dirichlet eta function $\eta(s) = (1 - 2^{1-s}) \zeta(s)$ (acb_dirichlet_eta).

Computes the Riemann xi function $\xi(s)$ (acb_dirichlet_xi).

Computes the Hardy (Riemann–Siegel) theta function $\theta(z)$ (acb_dirichlet_hardy_theta).

Computes the Hardy Z-function $Z(z)$ (acb_dirichlet_hardy_z).

Computes the $n$-th nontrivial zero of the Riemann zeta function with positive imaginary part, $n \ge 1$ (acb_dirichlet_zeta_zero).

Compute the error function $\operatorname{erf}(x)$ (arb_hypgeom_erf, acb_hypgeom_erf).

Compute the complementary error function $\operatorname{erfc}(x) = 1 - \operatorname{erf}(x)$, accurately also where it is small (arb_hypgeom_erfc, acb_hypgeom_erfc).

Compute the imaginary error function $\operatorname{erfi}(x) = -i \operatorname{erf}(ix)$ (arb_hypgeom_erfi, acb_hypgeom_erfi).

Computes the inverse error function on $(-1, 1)$ (arb_hypgeom_erfinv).

Computes the inverse complementary error function on $(0, 2)$ (arb_hypgeom_erfcinv).

Compute the Fresnel integral $S(x)$, normalised ($\int_0^x \sin(\pi t^2/2)\,dt$) if normalized is nonzero and $\int_0^x \sin(t^2)\,dt$ otherwise (arb_hypgeom_fresnel, acb_hypgeom_fresnel).

Compute the Fresnel integral $C(x)$, with the same normalisations as arb_fpwrap_double_fresnel_s.

Compute the upper incomplete gamma function $\Gamma(s, z)$ (arb_hypgeom_gamma_upper, acb_hypgeom_gamma_upper); regularized selects the variants as in those functions (0: plain, 1: divided by $\Gamma(s)$, 2: the generalised exponential integral normalisation).

Compute the lower incomplete gamma function $\gamma(s, z)$ (arb_hypgeom_gamma_lower, acb_hypgeom_gamma_lower), with regularized as for those functions.

Compute the incomplete beta function $B(a, b; z)$ (arb_hypgeom_beta_lower, acb_hypgeom_beta_lower), regularized if regularized is nonzero.

Compute the generalised exponential integral $E_s(z)$ (arb_hypgeom_expint, acb_hypgeom_expint).

Compute the exponential integral $\operatorname{Ei}(x)$ (arb_hypgeom_ei, acb_hypgeom_ei).

Compute the sine integral $\operatorname{Si}(x)$ (arb_hypgeom_si, acb_hypgeom_si).

Compute the cosine integral $\operatorname{Ci}(x)$ (arb_hypgeom_ci, acb_hypgeom_ci).

Compute the hyperbolic sine integral $\operatorname{Shi}(x)$ (arb_hypgeom_shi, acb_hypgeom_shi).

Compute the hyperbolic cosine integral $\operatorname{Chi}(x)$ (arb_hypgeom_chi, acb_hypgeom_chi).

Compute the logarithmic integral $\operatorname{li}(x)$, or the offset integral $\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)$ if offset is nonzero (arb_hypgeom_li, acb_hypgeom_li).

Compute the dilogarithm $\operatorname{Li}_2(x)$ (arb_hypgeom_dilog, acb_hypgeom_dilog).

Compute the Bessel function of the first kind $J_\nu(x)$ (arb_hypgeom_bessel_j, acb_hypgeom_bessel_j).

Compute the Bessel function of the second kind $Y_\nu(x)$ (arb_hypgeom_bessel_y, acb_hypgeom_bessel_y).

Compute the modified Bessel function of the first kind $I_\nu(x)$ (arb_hypgeom_bessel_i, acb_hypgeom_bessel_i).

Compute the modified Bessel function of the second kind $K_\nu(x)$ (arb_hypgeom_bessel_k, acb_hypgeom_bessel_k).

Compute $e^x K_\nu(x)$, which avoids underflow for large $x$ (arb_hypgeom_bessel_k_scaled, acb_hypgeom_bessel_k_scaled).

Compute the Airy function $\operatorname{Ai}(x)$ (arb_hypgeom_airy, acb_hypgeom_airy).

Compute $\operatorname{Ai}'(x)$.

Compute the Airy function $\operatorname{Bi}(x)$.

Compute $\operatorname{Bi}'(x)$.

Computes the $n$-th zero $a_n$ of $\operatorname{Ai}$, $n \ge 1$ (arb_hypgeom_airy_zero); the zeros are negative.

Computes the $n$-th zero $a'_n$ of $\operatorname{Ai}'$, $n \ge 1$.

Computes the $n$-th zero $b_n$ of $\operatorname{Bi}$, $n \ge 1$.

Computes the $n$-th zero $b'_n$ of $\operatorname{Bi}'$, $n \ge 1$.

Compute the regular Coulomb wave function $F_\ell(\eta, x)$ (arb_hypgeom_coulomb, acb_hypgeom_coulomb).

Compute the irregular Coulomb wave function $G_\ell(\eta, x)$.

Compute the complex Coulomb wave functions $H^{\pm}_\ell(\eta, z) = G_\ell \pm i F_\ell$ (acb_hypgeom_coulomb).

Compute the Chebyshev function of the first kind $T_n(x)$ for real or complex $n$ (arb_hypgeom_chebyshev_t, acb_hypgeom_chebyshev_t).

Compute the Chebyshev function of the second kind $U_n(x)$.

Compute the Jacobi function $P_n^{(a,b)}(x)$ (arb_hypgeom_jacobi_p, acb_hypgeom_jacobi_p). In FLINT 3.6 and earlier the real version computed $P_n^{(a,x)}(0)$ instead; the complex version was correct.

Compute the Gegenbauer function $C_n^{(m)}(x)$.

Compute the associated Laguerre function $L_n^{(m)}(x)$.

Compute the Hermite function $H_n(x)$.

Compute the Legendre function of the first kind $P_n^m(x)$; type selects the branch cut convention as in arb_hypgeom_legendre_p (0 or 1).

Compute the Legendre function of the second kind $Q_n^m(x)$, with type as in arb_hypgeom_legendre_q.

Sets res1 to the root of index $k$ (with $0 \le k < n$, in decreasing order) of the Legendre polynomial $P_n$ and res2 to the corresponding Gauss–Legendre quadrature weight (arb_hypgeom_legendre_p_ui_root). Fails if $k \ge n$.

Computes the spherical harmonic $Y_n^m(\theta, \phi)$ with $\theta$ = x1 and $\phi$ = x2 (acb_hypgeom_spherical_y).

Compute ${}_0F_1(;a;x)$, or its regularized version ${}_0F_1(;a;x)/\Gamma(a)$ if regularized is nonzero (arb_hypgeom_0f1, acb_hypgeom_0f1).

Compute the confluent hypergeometric function ${}_1F_1(a;b;x)$, regularized if regularized is nonzero (arb_hypgeom_1f1, acb_hypgeom_1f1).

Compute the confluent hypergeometric function $U(a, b, x)$ (arb_hypgeom_u, acb_hypgeom_u).

Compute the Gauss hypergeometric function ${}_2F_1(a, b; c; x)$, regularized if regularized is nonzero (arb_hypgeom_2f1, acb_hypgeom_2f1).

Compute the generalised hypergeometric function ${}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z)$ with the parameters given as arrays of length $p$ and $q$, regularized if regularized is nonzero (arb_hypgeom_pfq, acb_hypgeom_pfq).

Compute the arithmetic-geometric mean $M(x, y)$ (arb_agm, acb_agm).

Computes the complete elliptic integral of the first kind $K(m)$ in terms of the parameter $m = k^2$ (acb_elliptic_k).

Computes the complete elliptic integral of the second kind $E(m)$ (acb_elliptic_e).

Computes the complete elliptic integral of the third kind $\Pi(n, m)$ (acb_elliptic_pi).

Computes the incomplete elliptic integral of the first kind $F(\phi, m)$; if pi is nonzero, the argument is $\pi\phi$ instead of $\phi$ (acb_elliptic_f).

Computes the incomplete elliptic integral of the second kind $E(\phi, m)$, with pi as for arb_fpwrap_cdouble_elliptic_f (acb_elliptic_e_inc).

Computes the incomplete elliptic integral of the third kind $\Pi(n, \phi, m)$, with pi as above (acb_elliptic_pi_inc).

Computes the Carlson symmetric elliptic integral $R_F(x, y, z)$ (acb_elliptic_rf); option is passed as its flags and should be 0.

Computes the Carlson symmetric elliptic integral $R_G(x, y, z)$ (acb_elliptic_rg), with option as above.

Computes the Carlson symmetric elliptic integral $R_J(x, y, z, w)$ (acb_elliptic_rj), with option as above.

Computes the Weierstrass elliptic function $\wp(z, \tau)$ for the lattice with periods $1$ and $\tau$ (acb_elliptic_p).

Computes $\wp'(z, \tau)$ (acb_elliptic_p_prime).

Computes the inverse Weierstrass elliptic function $\wp^{-1}(z, \tau)$ (acb_elliptic_inv_p).

Computes the Weierstrass zeta function $\zeta(z, \tau)$ (acb_elliptic_zeta).

Computes the Weierstrass sigma function $\sigma(z, \tau)$ (acb_elliptic_sigma).

Computes the Jacobi theta function $\theta_1(z, \tau)$ in the convention of acb_modular_theta.

Computes $\theta_2(z, \tau)$.

Computes $\theta_3(z, \tau)$.

Computes $\theta_4(z, \tau)$.

Computes the Dedekind eta function $\eta(\tau)$ (acb_modular_eta).

Computes the modular $j$-invariant $j(\tau)$ (acb_modular_j).

Computes the modular lambda function $\lambda(\tau)$ (acb_modular_lambda).

Computes the modular discriminant $\Delta(\tau)$ (acb_modular_delta).