Abstract :
The double Zeta function of Barnes, ζ2(v, z,w), is considered for large and small values
of z and w with w > 0, |Arg(z)| < π and v = 1, 2. Two integral representations are
obtained for ζ2(v, z,w). These integrals define the analytical continuation of the double
Zeta function, primarily defined for (v) > 2 and (z) > 0, to the whole complex z-plane
and complex v-plane with |Arg(z)|<π and v = 1, 2. Six asymptotic expansions for large
and small w or z are derived from these integrals. The expansions are all accompanied by
error bounds at any order of the approximation. Numerical experiments show that these
bounds are very accurate for real values of the variables.
2002 Elsevier Science (USA). All rights reserved
