Euler Maclaurin Formula

Euler Maclaurin Formula

The Euler Maclaurin Formula in Mathematics is a way to calculate the difference between an integral and a closely comparable sum. It can be used to estimate finite sums and infinite series using integrals and techniques of calculus, or it can be used to calculate integrals using finite sums. For instance, the Euler Maclaurin Formula leads directly to Faulhaber’s formula for the sum of powers and many more asymptotic expansions. Around 1735, Colin Maclaurin and Leonhard Euler independently discovered this formula. While Maclaurin used it to compute integrals, Euler used it to compute slowly convergent infinite series. Later, it was extended to include Darboux’s formula.

The summation formula by Euler and Maclaurin is a crucial tool for numerical analysis. In other words, it provides us with an estimate of the total by using an integral with a Bernoulli number-based error term. It requires that the real function f have a continuous k-th derivative as the condition. The fractional portion of a real number is represented by the symbol x. The book [Ka] by Victor Kac provides a proof of this theorem using h-calculus. Here, one can know about numerous uses for this formula. In the first half of the XVIII century, Euler and Maclaurin independently and almost simultaneously discovered this formula. However, neither of them was able to get the remaining term which is the most crucial. Both employed an iterative technique to find Bernoulli’s numbers bi, but Euler’s strategy was solely analytical whereas Maclaurin’s was mostly based on geometric structure. Later, S.D. Poisson introduced the concept of the remainder.

One may get more historical information about this formula at [Mi]. With the exception of the first term, b1 = 1/ 2, all of the odd terms in the series are zero and rational. This fact can be demonstrated by taking into account the Taylor expansion of ex. As a result, it appears to be possible to eliminate the odd terms from the summations in the formulas. As one can see, the distribution of the first few of these numbers does not follow a clear pattern. But when n is really large, one is aware of an asymptotic expansion of the Bernoulli numbers, bn.

What is the Euler Maclaurin Formula

The Euler Maclaurin Formula makes it simple to evaluate the relationship between sums and integrals. The Euler Maclaurin Formula is regarded as a significant integral connection. An essential tool for calculating sums or integrals of functions is provided by Euler-Maclaurin. It allows one to deduce numerous crucial identities in the analysis of crucial functions like the Riemann zeta function, gamma function, or Euler constant.