zeta.c

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/*                                                     zeta.c
 *
 *     Riemann zeta function of two arguments
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, q, y, zeta();
 *
 * y = zeta( x, q );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 *
 *                 inf.
 *                  -        -x
 *   zeta(x,q)  =   >   (k+q)
 *                  -
 *                 k=0
 *
 * where x > 1 and q is not a negative integer or zero.
 * The Euler-Maclaurin summation formula is used to obtain
 * the expansion
 *
 *                n
 *                -       -x
 * zeta(x,q)  =   >  (k+q)
 *                -
 *               k=1
 *
 *           1-x                 inf.  B   x(x+1)...(x+2j)
 *      (n+q)           1         -     2j
 *  +  ---------  -  -------  +   >    --------------------
 *        x-1              x      -                   x+2j+1
 *                   2(n+q)      j=1       (2j)! (n+q)
 *
 * where the B2j are Bernoulli numbers.  Note that (see zetac.c)
 * zeta(x,1) = zetac(x) + 1.
 *
 *
 *
 * ACCURACY:
 *
 *
 *
 * REFERENCE:
 *
 * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
 * Series, and Products, p. 1073; Academic Press, 1980.
 *
 */

/*
 * Cephes Math Library Release 2.0:  April, 1987
 * Copyright 1984, 1987 by Stephen L. Moshier
 * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */
 
#include "mconf.h"
extern double MACHEP;
 
/* Expansion coefficients
 * for Euler-Maclaurin summation formula
 * (2k)! / B2k
 * where B2k are Bernoulli numbers
 */
static double A[] = {
    12.0,
    -720.0,
    30240.0,
    -1209600.0,
    47900160.0,
    -1.8924375803183791606e9,   /*1.307674368e12/691 */
    7.47242496e10,
    -2.950130727918164224e12,   /*1.067062284288e16/3617 */
    1.1646782814350067249e14,   /*5.109094217170944e18/43867 */
    -4.5979787224074726105e15,  /*8.028576626982912e20/174611 */
    1.8152105401943546773e17,   /*1.5511210043330985984e23/854513 */
    -7.1661652561756670113e18   /*1.6938241367317436694528e27/236364091 */
};
 
/* 30 Nov 86 -- error in third coefficient fixed */
 
 
double zeta(double x, double q)
{
    int i;
    double a, b, k, s, t, w;
 
    if (x == 1.0)
        goto retinf;
 
    if (x < 1.0) {
      domerr:
        sf_error("zeta", SF_ERROR_DOMAIN, NULL);
        return (NAN);
    }
 
    if (q <= 0.0) {
        if (q == floor(q)) {
            sf_error("zeta", SF_ERROR_SINGULAR, NULL);
          retinf:
            return (INFINITY);
        }
        if (x != floor(x))
            goto domerr;        /* because q^-x not defined */
    }
 
    /* Asymptotic expansion
     * https://dlmf.nist.gov/25.11#E43
     */
    if (q > 1e8) {
        return (1/(x - 1) + 1/(2*q)) * pow(q, 1 - x);
    }
 
    /* Euler-Maclaurin summation formula */
 
    /* Permit negative q but continue sum until n+q > +9 .
     * This case should be handled by a reflection formula.
     * If q<0 and x is an integer, there is a relation to
     * the polyGamma function.
     */
    s = pow(q, -x);
    a = q;
    i = 0;
    b = 0.0;
    while ((i < 9) || (a <= 9.0)) {
        i += 1;
        a += 1.0;
        b = pow(a, -x);
        s += b;
        if (fabs(b / s) < MACHEP)
            goto done;
    }
 
    w = a;
    s += b * w / (x - 1.0);
    s -= 0.5 * b;
    a = 1.0;
    k = 0.0;
    for (i = 0; i < 12; i++) {
        a *= x + k;
        b /= w;
        t = a * b / A[i];
        s = s + t;
        t = fabs(t / s);
        if (t < MACHEP)
            goto done;
        k += 1.0;
        a *= x + k;
        b /= w;
        k += 1.0;
    }
done:
    return (s);
}