ABSTRACT
This chapter describes and applies some basic techniques for the construction of analytical continuations of zeta functions. It mentions contour integral representations and Mellin transformations as the most important ingredients. These techniques obtain the zeta function (for a specific class of examples) for all required values of the complex parameter. The chapter explains the case of a massive scalar field on the three-dimensional ball, where the field is supposed to fulfill Dirichlet boundary conditions at the boundary, and the D-dimensional generalized cone. It reduces the analysis of the zeta function on the generalized cone to the one on a base manifold, which for specific manifolds can be given very explicitly in terms of known and well-studied zeta functions. As a result residues, function values and derivatives at whatever values of needed can be calculated in the way it was possible for the case of the three-dimensional ball.
