Asymptotic expansions of the double Zeta function

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