The Euler–Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler–Maclaurin summation formula and the Poisson summation formula as special cases. By making use of the modified formula, a possible numerical summation method is obtained and the remainder can be controlled. The modified formula is also generalized from one dimension to two dimensions. Approximate expressions of partition functions of a classical particle in square well in 1D and 2D and that of a quantum rotator are obtained with error estimation.

中文翻译：

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改进的一维和二维欧拉-麦克劳林公式在统计物理中的应用

欧拉-麦克劳林求和公式通过将周期伯努利多项式展开为其傅立叶级数并进行切割，将欧拉-麦克劳林求和公式推广为修改形式，其中包括欧拉-麦克劳林求和公式和泊松求和公式作为特例。利用修改后的公式，得到了一种可能的数值求和方法，并可以控制余数。修改后的公式也从一维推广到二维。用误差估计得到了方阱中经典粒子在一维和二维中的配分函数和量子旋转器配分函数的近似表达式。