| Pré. | Proc. |
zeta
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 zeta(x,1) = zetac(x) + 1.
(see zetac)
ACCURACY:
REFERENCE:
Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Products, p. 1073; Academic Press, 1980.