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.

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}.

Compute \operatorname{Si}(z) with N terms of the asymptotic expansion, or of the {}_1F_2 series summed at working precision wp and rounded to prec, including a bound for the truncation error. These are the two methods of arb_hypgeom_si.

Compute \operatorname{Ci}(z) with N terms of the asymptotic expansion or of the {}_2F_3 series, including a bound for the truncation error. These are the two methods of arb_hypgeom_ci.

Sets dp to an upper bound for |P'_n(x)| and dp2 to an upper bound for |P''_n(x)|, for x representing real numbers with |x| \le 1, given the precomputed x2sub1 = 1 - x^2 (or x^2 - 1). Used to bound the propagated error in arb_hypgeom_legendre_p_ui.