Euler-Maclaurin formula

(redirected from Euler's summation formula)

Euler-Maclaurin formula

[¦ȯi·lər mə′klȯr·ən ‚fȯr·myə·lə]
(mathematics)
A formula used in the numerical evaluation of integrals, which states that the value of an integral is equal to the sum of the value given by the trapezoidal rule and a series of terms involving the odd-numbered derivatives of the function at the end points of the interval over which the integral is evaluated.
References in periodicals archive ?
The following important result, known as Euler's summation formula, gives an exact formula for the difference between such a sum and the corresponding integral.
Then from the Euler's summation formula (see reference [2]), we may immediately deduce that
From the Euler's summation formula, we obtain the main term of lnm
From the definition of a(n), the Euler's summation formula (see [4]) and the properties of the Mobius function, we can get
And then using the Euler's summation formula we have
Taking f(t) = ln t in Euler's summation formula [2] we obtain: