  This paper describes a trapezoidal quadrature method for the discretization
of weakly singular, singular and hypersingular boundary integral operators with
complex symmetric quadratic forms. Such integral operators naturally arise when
complex coordinate methods or complexified contour methods are used for the
solution of time-harmonic acoustic and electromagnetic interface problems in
three dimensions. The quadrature is an extension of a locally corrected
punctured trapezoidal rule in parameter space wherein the correction weights
are determined by fitting moments of error in the punctured trapezoidal rule,
which is known analytically in terms of the Epstein zeta function. In this
work, we analyze the analytic continuation of the Epstein zeta function and the
generalized Wigner limits to complex quadratic forms; this analysis is
essential to apply the fitting procedure for computing the correction weights.
We illustrate the high-order convergence of this approach through several
numerical examples.
