The Euler–MacLaurin Formula

Manuel Eberl 🌐

March 10, 2017

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

The Euler-MacLaurin formula relates the value of a discrete sum to that of the corresponding integral in terms of the derivatives at the borders of the summation and a remainder term. Since the remainder term is often very small as the summation bounds grow, this can be used to compute asymptotic expansions for sums.

This entry contains a proof of this formula for functions from the reals to an arbitrary Banach space. Two variants of the formula are given: the standard textbook version and a variant outlined in Concrete Mathematics that is more useful for deriving asymptotic estimates.

As example applications, we use that formula to derive the full asymptotic expansion of the harmonic numbers and the sum of inverse squares.

BSD License

Topics

Theories of Euler_MacLaurin