The SE and DE formulas are known as efficient quadrature formulas for integrals with endpoint singularities. Especially for integrals with algebraic singularity, explicit error bounds in a computable form have been given, which are useful for computation with guaranteed accuracy. Such explicit error bounds have also given for integrals with logarithmic singularity. However, the 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 remedy these points, 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 may 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性积分的有效求积公式。特别地，对于具有代数奇异性的积分，已有可计算形式的显式误差界被提出，这对保证精度的计算十分有用。对于具有对数奇异性的积分，此类显式误差界也已给出。然而，现有误差界存在两点值得商榷之处。第一点涉及对数奇异性发散速度的高估。第二点涉及同时存在对数和代数奇异性的情况。为改进这两点，本研究为具有对数和代数奇异性的积分提供了新的误差界。尽管上述现有及新的误差界处理的是有限区间上的积分，但SE与DE公式也可应用于半无限区间上的积分。基于新结果，本研究进一步为原点处具有对数和代数奇异性的半无限区间积分提供了新的误差界。