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.

翻译：本文描述了一种用于离散化具有复对称二次形式的弱奇异、奇异和超奇异边界积分算子的梯形求积方法。当使用复坐标法或复化围道法求解三维时间调和声学和电磁界面问题时，此类积分算子自然出现。该求积方法是参数空间中局部校正穿孔梯形规则的推广，其中校正权重通过拟合穿孔梯形规则的误差矩确定，该误差矩可通过Epstein ζ函数解析表示。本研究分析了Epstein ζ函数到复二次形式的解析延拓以及广义Wigner极限；这一分析对于应用拟合过程计算校正权重至关重要。通过多个数值算例，我们验证了该方法的高阶收敛性。