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The Euler-Maclaurin summation formula is generalized to a modified form byexpanding the periodic Bernoulli polynomials as its Fourier series and takingcuts, which includes both the Euler-Maclaurin summation formula and thePoission summation formula as special cases. By making use of the modifiedformula, a numerical summation method is obtained and the error can becontrolled. The modified formula is also generalized from one dimention to twodimentions. Examples of its applications in statistical physics are alsodiscussed.

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