Page "Euler–Maclaurin formula" Paragraph 102

The Euler – MacLaurin summation formula can thus be seen to be an outcome of the representation of functions on the unit interval by the direct product of the Bernoulli polynomials and their duals.

Note, however, that the representation is not complete on the set of square-integrable functions.

The expansion in terms of the Bernoulli polynomials has a non-trivial kernel.

In particular, sin ( 2πnx ) lies in the kernel ; the integral of sin ( 2πnx ) is vanishing on the unit interval, as is the difference of its derivatives at the endpoints.

This is the essentially the reason for the restriction to exponential type of less than 2π: the function sin ( 2πnz ) grows faster than e < sup > 2π | z |</ sup > along the imaginary axis!

Essentially, Euler-MacLaurin summation can be applied whenever Carlson's theorem holds ; the Euler-MacLaurin formula is essentially a result obtaining from the study of finite differences and Newton series.