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

Sets res to the Jacobi polynomial (or Jacobi function)

For nonnegative integer n this is a polynomial in a, b and z, also for parameters where the hypergeometric series is undefined; it is then evaluated by direct methods.

Sets res to the Gegenbauer polynomial (or Gegenbauer function)

For nonnegative integer n this is a polynomial in m and z, also where the hypergeometric series is undefined; it is then evaluated by direct methods.

Sets res to the Laguerre polynomial (or Laguerre function)

For nonnegative integer n this is a polynomial in m and z, also where the hypergeometric series is undefined; it is then evaluated by direct methods.

Sets res to the Hermite polynomial (or Hermite function)

Sets res to the associated Legendre function of the first kind of degree n, order m and argument z; for m = 0 this is the Legendre polynomial P_n(z). Several branch cut conventions are in use. If type is 0, the function is

with cuts (-\infty, -1] and [1, \infty), and if type is 1, the function is

with the cut (-\infty, 1]. Type 0 and type 1 correspond to type 2 and type 3 in Mathematica and mpmath; other values of type raise an exception.

Sets res to the associated Legendre function of the second kind of degree n, order m and argument z; for m = 0 this is the Legendre function Q_n(z). If type is 0, the function is

and if type is 1,

Type 0 and type 1 correspond to type 2 and type 3 in Mathematica and mpmath; other values of type raise an exception.

For nonnegative integers n and m, sets res to (1-z^2)^{-m/2} P_n^m(z), which is a polynomial in z, computed with the three-term recurrence in the degree starting from P_m^m. The result is zero for m > n, and indeterminate if (n - m)/4 > \mathit{prec}, where the recurrence would lose all accuracy.

Sets res to the spherical harmonic of degree n and order m at latitude angle \theta and longitude angle \phi, normalized so that

The definition is extended to negative m and n by symmetry. The function is a polynomial in \cos\theta and \sin\theta and is evaluated with acb_hypgeom_legendre_p_uiui_rec.

Sets res to \operatorname{Li}_2(z) computed with the series expansion in w = \log(z), which converges like |w/(2\pi)|^n. This gives good convergence near z = e^{\pm i \pi / 3}, where the hypergeometric series fail. Since the coefficients involve Bernoulli numbers, the method is meant for moderate precision.

Sets res to \operatorname{Li}_2(z) computed with the hypergeometric series at 0, which is efficient only for |z| \ll 1.

Sets res to \operatorname{Li}_2(z) for z close to 0, using the Taylor series below 40000 bits (or for tiny |z|) and the bit-burst algorithm otherwise.

Sets res to \operatorname{Li}_2(z) computed by one of the transformations 1/z, 1-z, z/(z-1), 1/(1-z), selected by algorithm from 1 to 4, followed by acb_hypgeom_dilog_zero at the transformed argument. For algorithm between 5 and 7, it instead starts from the respective point \pm i, (1\pm i)/2 or 1 \pm i (with the sign of the imaginary part of z), where \operatorname{Li}_2 has a closed form in terms of \pi, \log 2 and Catalan's constant, and continues to z with the bit-burst method; these require that the imaginary part of z does not contain 0. The result is indeterminate if the transformation is not valid on the whole input ball.

Sets res to \operatorname{Li}_2(z) - \operatorname{Li}_2(a) computed with the Taylor expansion at a, summed by binary splitting. Both a and z should be well isolated from 0 and 1, except that a may be exactly 0. If the segment from a to z crosses the branch cut, the result is the analytic continuation along the segment rather than the principal branch.

Sets z0 to a point with a short binary expansion close to z and res to \operatorname{Li}_2(z) - \operatorname{Li}_2(z_0), computed with the bit-burst algorithm: z_0 is z truncated to 16 fractional bits, and continuation steps lead from it through the truncations of z to 32, 64, 128, ... bits to z. If z is close to the branch cut, it sets z0 to z and res to 0.

Sets res to the dilogarithm \operatorname{Li}_2(z) with an automatic choice of method: the series at 0 after the transformation that minimizes |z|, the Bernoulli series near e^{\pm i\pi/3} below 10000 bits, and expansions at the points of acb_hypgeom_dilog_transform otherwise.

Sets C to a factor such that \left|\sum_{k=n}^{\infty} T(k)\right| \le C |T(n)|. The bound is infinite if p > q, if \operatorname{Re}(b_i + n) > 0 fails for some i, or if n is too small for the term ratio to be bounded below 1, even when the series converges. See Convergent series.

Heuristically chooses the number of terms n for summing the series at prec bits of working precision.

Like acb_hypgeom_pfq_choose_n for power series parameters and argument, using their constant terms; the result is at least len.

Sets s = \sum_{k=0}^{n-1} T(k) and t = T(n). Requires n \ge 0. The outputs must not alias the inputs. No truncation error is added.

Like acb_hypgeom_pfq_sum_bs and acb_hypgeom_pfq_sum, but given w = 1/z (and also z for the second function), which is advantageous when w has few bits, as in asymptotic expansions at an integer or short argument: binary splitting then divides by w instead of multiplying by the long z.

Sets res to

computed directly from the series with acb_hypgeom_pfq_sum and a rigorous bound for the truncation error \varepsilon from acb_hypgeom_pfq_bound_factor. If n < 0, the number of terms is chosen with acb_hypgeom_pfq_choose_n. If T(n) is exactly zero (a terminating series), no error is added. For real input the error is added to the real part only.

Sets s = \sum_{k=0}^{n-1} T(k) and t = T(n) for parameters and argument that are power series, with all arithmetic truncated to length len. Requires n \ge 0 and \mathit{len} \ge 1. The outputs must not alias the inputs.

Sets res to {}_pf_{q}(z) (regularized if regularized is set) for parameters and argument that are power series, truncated to length len \ge 1, summing n terms of the series and adding a rigorous bound for the truncation error in each coefficient (see Power series parameters). If n < 0, the number of terms is chosen automatically.