  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
Poission summation formula as special cases. By making use of the modified
formula, a numerical summation method is obtained and the error can be
controlled. The modified formula is also generalized from one dimention to two
dimentions. Examples of its applications in statistical physics are also
discussed.
