Euler–Maclaurin formula

The Euler - Maclaurin formula or Euler's summation formula (after Leonhard Euler and Colin Maclaurin ) is a mathematical formula for calculating a sum of function values ​​by the values ​​of the derivatives of this function to the summation limits - as is Euler encountered them. In a modified form it allows for the numerical approximation of a definite integral of individual values ​​of the integrand and its derivatives - as it drew Maclaurin.

Notation Note

For a sufficiently often differentiable function of one variable is the entire article for all the spelling of a shorthand notation for

The -th derivative from to, evaluated at

Euler - Maclaurin formula for Integralapproximation

Is (that is, a function which is at least twice continuously differentiable on the interval ) was added. Then there exists a number such that

Holds, where the Bernoulli numbers.

This is a simple form of Euler Maclaurin empirical formula, wherein said summation comprises only two terms ( with index 0 and 1). The term is exactly the approximation of an integral by the area of ​​a trapezoid. The following sum provides a correction term and the last term the error that appears with it. Hence the name of this formula in the numerical integration theory also " trapezoidal rule with final correction ". Using this formula, it is only possible to determine the error of the trapezoidal rule for the interval, if known. Thus, this formula, although no estimate, but an equality is, however, only in the form of an existence statement.

Euler - Maclaurin formula for Summenapproximation

The usual version of the above sum formula with effective residual limb specification is obtained by being surrounded on

And then the function is replaced by a function that is applied in any one interval endpoints, but the remainder calculated explicitly as a function of the " next " derivative. To do this, simply sums this formula applied to a corresponding number of shifted unit intervals, which cover the given interval of exactly on. Be and at least twice continuously differentiable on, then one obtains

In which

With the Bernoulli's polynomials. This is the Euler formula for sum of Maclaurin series for the determination, wherein is already sufficient. We also used the convention

For the -th derivative, then the formula can be much more elegant to

Rewrite - you do not stop at an even index the summation to make a residual limb determination - with the only Bernoulli number is not 0 with odd index. If now even the border crossing performed, one obtains

For practical use. However, it should be noted that this is often not convergent, but only an asymptotic series, more precisely an expansion in derivatives of the function represents.

It also uses the so-called Bernoulli numbers of the second kind, and for all other indexes - note for all odd, so can the above equation in a more symmetrical form rewrite:

Applications

• The classic problem of determining the power sums of the first natural numbers can now simply means umtransformieren to:

• Another classic example is the choice, which gives the general ( logarithmic ) Stirling number of the summation formula and so the faculties even for very large arguments quickly, or for non-integer arguments can compute efficiently.
• One application of numerics is opened, if one changes the Euler - Maclaurin formula as its integral:

• Is used in place of the trapezoidal rule the midpoint rule, to thus replace the summation of the function values ​​by one can avoid the sometimes problematic function evaluation at the edges. This is particularly the case if the integrand is not defined on the edge numerically unstable ( for example ), or ( for example ). Here, the differences of the odd derivatives are reduced by a factor of. The contributions to the total error of the differences are so small as to be expected when using the midpoint rule. The factor is found similarly in the Romberg integration of even and odd functions again. It is considered that the differences of the derivative is also available from the application of the center rule, the integral edges.
• An important application is the Euler - Maclaurin formula for periodic functions that are to be integrated over one or more periods. For such functions, all derivatives at the integral limits are identically equal and therefore vanish there (also) the sum of the differences of the ( odd ) derivatives. The integral can be approximated by so - fold application of the trapezoidal rule with an error of order. This explains, among other things, why the discrete Fourier transform by means of Chebyshev polynomials have by summation and the approximation of such a high accuracy. It should be noted that the discrete Fourier transformation is usually related to the Euler formula Maclaurin trapezoidal rule during the approximation with Chebyshev polynomials using the midpoint rule. For applications but you can also work with the other summation rule. The equivalence is proved by the Euler - Maclaurin formula.
• The Euler - Maclaurin formula allows also an important application with functions that can be mirrored on both integral boundaries so that they can be continued steadily along with all dissipation. For such functions are all the odd derivatives of the integral limits zero, and therefore the sum of the differences of the odd derivatives also disappears. Consequently, here the error is of the order is independent of the theoretical backgrounds of the Gaussian quadrature can be the Gauss - Chebyshev integration and the integral derived solely on the Euler - Maclaurin formula.