Ability to evaluate contour integrals is central to both the theory and the utilization of analytic functions. We present here a complex plane realization of the Euler–Maclaurin formula that includes weights also at some grid points adjacent to each end of a line segment (made up of equispaced grid points, along which we use the trapezoidal rule). For example, with a |$5\times 5$| ‘correction stencil’ (with weights about two orders of magnitude smaller than those of the trapezoidal rule), the accuracy is increased from |$2$|nd to |$26$|th order.

中文翻译：

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在复杂平面上的网格上给出的解析函数的轮廓积分

评估轮廓积分的能力对于理论和解析函数的使用都是至关重要的。在这里，我们介绍了Euler-Maclaurin公式的复杂平面实现，该公式还包括在与线段两端相邻的某些网格点处的权重（由等距网格点组成，我们使用梯形法则）。例如，使用| $ 5 \ times 5 $ | “校正模具”（权重比梯形法则小两个数量级），准确性从| $ 2 $ | nd || $ 26 $ | 顺序。