acb_hypgeom.h

Sets res to the rising factorial $(x)_n = x(x+1)\cdots(x+n-1)$; acb_hypgeom_rising allows any complex $n$ and computes $\Gamma(x+n)/\Gamma(x)$.

Sets the vector res of length len to the coefficients of the polynomial $(x + t)_n \in \mathbb{C}[t]$, truncated if $\mathit{len} < n+1$ and padded with zeros if $\mathit{len} > n+1$.

Sets res to the log-rising factorial $\log\,(x)_n = \sum_{k=0}^{n-1} \log(x+k)$, the sum of principal logarithms, which differs from the principal logarithm of $(x)_n$ by a multiple of $2\pi i$.

Sets the vector res to the coefficients of $\log\,(x+t)_n$ as a power series in $t$, truncated to length len, with the constant term as in acb_hypgeom_log_rising_ui.

Sets res to the gamma function $\Gamma(x)$. The result is indeterminate if $x$ contains a pole.

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

Sets res to the principal branch of the log-gamma function $\log \Gamma(x)$, which is analytic except on the branch cut $(-\infty, 0]$ and differs from $\log(\Gamma(x))$ by a multiple of $2\pi i$.

Sets s to $\sum_{n=1}^{N-1} B_{2n} / (2n(2n-1) z^{2n-1})$, the sum in the Stirling series truncated before the term of index $N$, using Horner's scheme or the improved algorithm with splitting parameter $K$ (0 to choose it), as arb_hypgeom_gamma_stirling_sum_horner and arb_hypgeom_gamma_stirling_sum_improved do for real $z$. 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 by a rising factorial and, for real part below $-5$, the reflection formula. See arb_hypgeom_gamma_stirling and The Stirling series.

Attempts to compute $\Gamma(x)$, or $1/\Gamma(x)$ if reciprocal is nonzero, with the tabulated Taylor series of $1/\Gamma(1+x)$ and argument reduction. Returns 1 on success and 0, without changing res, if $x$ or prec is outside the supported range.

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.

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. acb_hypgeom_1f1 is the same function as acb_hypgeom_m.

Sets res to $U^{*}(a,b,z)$ computed with $n$ terms of the asymptotic series $U^{*}(a,b,z) \sim {}_2F_0(a, a-b+1; ; -1/z)$, with a rigorous bound for the truncation error. Requires $n \ge 0$; $n < 0$ chooses the number of terms automatically.

Returns nonzero if the asymptotic series is expected to give $U(a,b,z)$ to prec bits, namely if $|z| > \mathit{prec} \log 2$ or $|z| > 2^{64}$ (assuming that $a$ and $b$ are small).

Sets res to $U(a,b,z)$ as a power series truncated to length len, for power series $a, b, z \in \mathbb{C}[[x]]$, using the connection formula with two ${}_1F_1$ series. If the constant term of $b$ is an integer, one extra derivative is computed and the removable singularity is divided out; it is then assumed that the coefficient of $x$ in $b$ is nonzero. The output is indeterminate if the constant term of $b$ is inexact and contains an integer.

Sets res to $U(a,b,z)$ computed from the convergent series by the connection formula

If $b$ is an integer the limit is computed with acb_hypgeom_u_1f1_series; the output is indeterminate if $b$ is inexact and contains an integer.

Sets res to $U(a,b,z)$.

Sets res to the confluent hypergeometric limit function ${}_0F_1(a, z)$, or ${}_0F_1(a,z)/\Gamma(a)$ if regularized is nonzero, using asymptotic expansions, direct summation, or an automatic choice.

Sets res to $\operatorname{erf}(z)$ computed respectively from

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

Sets re and im to upper bounds for the errors in the real and imaginary parts that result from replacing $\operatorname{erf}(z)$ by $\operatorname{erf}$ evaluated at the midpoint of $z$, using a bound for the first derivative $\frac{2}{\sqrt\pi} e^{-z^2}$ on the ball.

Sets res to the error function of the power series $z$, truncated to length len.

Sets res to the complementary error function $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$, avoiding the catastrophic cancellation of the subtraction when $\operatorname{Re} z$ is large and positive.

Sets res to the complementary error function of the power series $z$, truncated to length len.

Sets res to the imaginary error function $\operatorname{erfi}(z) = -i\operatorname{erf}(iz)$, computed with acb_hypgeom_erf.

Sets res to the imaginary error function 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, the integrands are $\sin(\tfrac12 \pi t^2)$ and $\cos(\tfrac12 \pi t^2)$, the convention of Abramowitz and Stegun and of the DLMF. The values are computed from $\operatorname{erf}((1 \pm i) w)$, where $w$ is $z$ times a real constant, evaluated at the midpoint of $z$ with a bound for the propagated error.

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 the generalized exponential integral $z^{-s} \Gamma(s,z) = E_{1-s}(z)$ if regularized is 2 (mainly for internal use; acb_hypgeom_expint is the interface for $E_s$).

Sets res to the upper incomplete gamma function of the constant $s$ and the power series $z$, truncated to length $n$, with regularized as in acb_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 $s$ and $z$, if regularized is 2.

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 acb_hypgeom_gamma_lower.

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

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

Sets res to the generalized exponential integral $E_s(z) = z^{s-1} \Gamma(1-s, z)$, computed with acb_hypgeom_gamma_upper.

Sets res to the exponential integral $\operatorname{Ei}(z)$, computed respectively from

or by an automatic choice based on the expected accuracy of the two methods. The result is indeterminate if $z$ contains zero.

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

Sets res to the sine integral $\operatorname{Si}(z)$, computed respectively from

or by an automatic choice. Real input is passed to arb_hypgeom_si.

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

Sets res to the cosine integral $\operatorname{Ci}(z)$, computed respectively from

or by an automatic choice. Positive real input is passed to arb_hypgeom_ci.

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

Sets res to the hyperbolic sine integral $\operatorname{Shi}(z) = -i \operatorname{Si}(iz)$, computed with acb_hypgeom_si.

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

Sets res to the hyperbolic cosine integral $\operatorname{Chi}(z)$, computed respectively from

or by an automatic choice.

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.

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 $J_\nu(z)$ computed with the asymptotic expansion of acb_hypgeom_u_asymp.

Sets res to $J_\nu(z)$ computed from the convergent series

using $J_{-n}(z) = (-1)^n J_n(z)$ for negative integer $\nu$.

Sets res to the Bessel function of the first kind $J_\nu(z)$.

Sets res to an upper bound, possibly crude, for $|J_\nu^{(d)}(z)|$ ($|J_\nu(z)|$ for $d = 0$). Implemented only for integer $\nu$, using for example $|J_n^{(d)}(z)| \le e^{|\operatorname{Im} z|}$; for non-integer $\nu$ the bound is infinite.

Sets res to $J_\nu(z)$ for the constant $\nu$ and the power series $z$, truncated to length len. Both functions call the generic _gr_poly_bessel_j_series with a complex ball context; as described for arb_hypgeom_bessel_j_series, the result is indeterminate if the constant term of $z$ is an inexact ball containing zero.

Sets res to the Bessel function of the second kind

or for $\nu = n$ an integer the limit

The output is indeterminate if $\nu$ is inexact and contains an integer.

Sets res1 to $J_\nu(z)$ and res2 to $Y_\nu(z)$, computed together. The Hankel functions $H_\nu^{(1)}(z), H_\nu^{(2)}(z) = J_\nu(z) \pm i Y_\nu(z)$ can be formed from them.

Sets res to the modified Bessel function of the first kind $I_\nu(z) = z^\nu (iz)^{-\nu} J_\nu(iz)$, computed respectively with the asymptotic expansion (see acb_hypgeom_bessel_j_asymp), the convergent series

or an automatic choice. acb_hypgeom_bessel_i_scaled computes $e^{-z} I_\nu(z)$; the asymp and 0f1 versions compute the scaled function if scaled is nonzero. For real arguments with an inaccurate result, the automatic versions also try arb_hypgeom_bessel_i_integration.

Sets res to the modified Bessel function of the second kind

valid for all $\nu$ and $z \ne 0$, using acb_hypgeom_u_asymp; or $e^z K_\nu(z)$ if scaled is nonzero.

Sets res to $K_\nu(z)$ (or $e^z K_\nu(z)$ if scaled is nonzero) as a power series truncated to length len, for power series $\nu, z \in \mathbb{C}[[x]]$, using

If the constant term of $\nu$ is an integer, one extra derivative is computed and the singularity is removed (the coefficient of $x$ in $\nu$ must then be nonzero). The output is indeterminate if the constant term of $\nu$ is inexact and contains an integer.

Sets res to $K_\nu(z)$ (or $e^z K_\nu(z)$ if scaled is nonzero) computed from

for $\nu \notin \mathbb{Z}$, and the limit computed with acb_hypgeom_bessel_k_0f1_series for $\nu \in \mathbb{Z}$. The output is indeterminate if $\nu$ is inexact and contains an integer.

Sets res to the modified Bessel function of the second kind $K_\nu(z)$, or to $e^z K_\nu(z)$, choosing between the asymptotic expansion and the convergent series. For real arguments with an inaccurate result they also try arb_hypgeom_bessel_k_integration.

Computes the Airy functions with the convergent series at the origin truncated after $n$ terms, including a bound for the truncation error.

Computes the Airy functions with the asymptotic expansions truncated after $n$ terms, including a bound for the truncation error. The bounds and connection formulas are given in Asymptotic expansions.

Sets the outputs to upper bounds for $|\operatorname{Ai}(z)|$, $|\operatorname{Ai}'(z)|$, $|\operatorname{Bi}(z)|$, $|\operatorname{Bi}'(z)|$ on the ball $z$, from first-order asymptotic expansions with error bounds. This is faster than calling acb_hypgeom_airy_asymp with $n = 1$.

Computes the Airy functions with an automatic choice of method.

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. The expansions of the derivatives are obtained by computing one more coefficient and applying _acb_poly_derivative.

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 F, G, Hpos, Hneg to the respective Coulomb wave functions. Any of the outputs may be NULL.

Sets F, G, Hpos, Hneg to the Taylor coefficients of the Coulomb wave functions at the point $z$, truncated to length len, using the differential equation to obtain the higher coefficients. Any of the outputs may be NULL.

Sets the outputs to the Coulomb wave functions evaluated at the power series $z$, truncated to length len. Any of the outputs may be NULL.

Sets res to $F(z)$, or $\mathbf{F}(z)$ if flags contains ACB_HYPGEOM_2F1_REGULARIZED, with an automatic choice of algorithm. The other flags declare that a parameter difference is an exact integer even if the input balls are inexact, so that the correct limit is computed where a transformation formula is singular.

Sets res to $F(z)$ (or $\mathbf{F}(z)$) by direct summation of the hypergeometric series, which converges for $|z| < 1$.

Sets res to $F(z)$ (or $\mathbf{F}(z)$) for power series parameters and argument, truncated to length len, by direct summation of the hypergeometric series.

Sets res to $F(z)$ (or $\mathbf{F}(z)$) computed with the linear fractional transformation selected by which: 1 for $z/(z-1)$, 2 for $1/z$, 3 for $1/(1-z)$, 4 for $1-z$ and 5 for $1-1/z$.

Sets res to $F(z)$ (or $\mathbf{F}(z)$) for $z$ near the points $\exp(\pm \pi i/3)$, where none of the transformations gives a convergent series, by analytic continuation along a path from a point where the direct series converges.

Given $F(z_0)$ and $F'(z_0)$ in f0 and f1, sets res0 and res1 to $F(z_1)$ and $F'(z_1)$, computed by integrating the hypergeometric differential equation along the straight segment from $z_0$ to $z_1$ with Taylor series and rigorous bounds. The points should be well isolated from the singular points 0 and 1. If $F$ denotes the regularized function, $f_0, f_1$ must be values of the regularized function.

Chooses the method for computing ${}_2F_1$ from the location of $z$: returns 0 for direct summation, 1 to 5 for the transformation with that index in acb_hypgeom_2f1_transform, or 6 for the corner case.

Sets res to the Chebyshev polynomial or Chebyshev function of the first or second kind