A modified Euler–Maclaurin formula in 1D and 2D with applications in statistical physics

Abstract

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.