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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(x, q)
double x, q;
{
int i;
double a, b, k, s, t, w;
if (x == 1.0)
goto retinf;
if (x < 1.0) {
domerr:
mtherr("zeta", DOMAIN);
return (NPY_NAN);
}
if (q <= 0.0) {
if (q == floor(q)) {
mtherr("zeta", SING);
retinf:
return (NPY_INFINITY);
}
if (x != floor(x))
goto domerr; /* because q^-x not defined */
}
/* Asymptotic expansion
* http://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);
}
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