A extension of the Euler-Maclaurin (E-M) formula to near-singular functions is presented. This extension is derived based on earlier generalized E-M formulas for singular functions. The new E-M formulas consists of two components: a ``singular'' component that is a continuous extension of the earlier singular E-M formulas, and a ``jump'' component associated with the discontinuity of the integral with respect to a parameter that controls near singularity. The singular component of the new E-M formulas is an asymptotic series whose coefficients depend on the Hurwitz zeta function or the digamma function. Numerical examples of near-singular quadrature based on the extended E-M formula are presented, where accuracies of machine precision are achieved insensitive to the strength of the near singularity and with a very small number of quadrature nodes.

翻译：本文提出了欧拉-麦克劳林（E-M）公式在近奇异函数上的一个推广形式。该推广基于早期针对奇异函数的广义E-M公式推导得出。新的E-M公式包含两个部分：其一是作为早期奇异E-M公式连续延拓的“奇异”分量，其二是与积分相对于控制近奇异性的参数不连续性相关的“跳跃”分量。新E-M公式的奇异分量是一个渐近级数，其系数依赖于赫尔维茨ζ函数或双伽玛函数。文中给出了基于推广E-M公式的近奇异求积分数值算例，这些算例在近奇异性强度变化时仍能保持机器精度级别的计算精度，且所需积分节点数极少。