The SE and DE formulas are known as efficient quadrature formulas for integrals with endpoint singularity. Particularly, for integrals with algebraic singularity, explicit error bounds in a computable form have been provided, which are useful for computations with guaranteed accuracy. Such explicit error bounds have also been provided for integrals with logarithmic singularity. However, these error bounds have two points to be discussed. The first point is on overestimation of divergence speed of logarithmic singularity. The second point is on the case where there exist both logarithmic and algebraic singularity. To address these issues, this study provides new error bounds for integrals with logarithmic and algebraic singularity. Although existing and new error bounds described above handle integrals over the finite interval, the SE and DE formulas can be applied to integrals over the semi-infinite interval. On the basis of the new results, this study provides new error bounds for integrals over the semi-infinite interval with logarithmic and algebraic singularity at the origin.

翻译：SE与DE公式是针对端点奇异积分的高效数值积分公式。特别地，对于具有代数奇异性的积分，已有研究提供了可计算形式的显式误差界，这对保证精度的计算具有重要意义。对于含对数奇异性的积分，此类显式误差界也已建立。然而，现有误差界存在两点值得商榷：其一在于对数奇异性发散速度的高估问题；其二涉及对数和代数奇异性共存的情形。为应对这些问题，本研究提出了针对含对数和代数奇异性积分的新误差界。尽管上述现有及新误差界处理的是有限区间上的积分，但SE与DE公式同样适用于半无限区间上的积分。基于新研究成果，本研究进一步给出了原点处含对数和代数奇异性的半无限区间积分的误差界。