In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Calculus is an important branch of mathematics. ... Euler redirects here. ... Colin Maclaurin Colin Maclaurin (February, 1698 - June 14, 1746) was a Scottish mathematician. ... Events April 16 - The London premiere of Alcina by George Frideric Handel, his first the first Italian opera for the Royal Opera House at Covent Garden. ...

The formula

If n is a natural number and f(x) is a smooth (meaning: sufficiently often differentiable) function defined for all real numbers x between 0 and n, then the integral In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a derivative is the rate of change of a quantity (e. ... Partial plot of a function f. ... In mathematics, the real numbers may be described informally in several different ways. ...

can be approximated by the sum

(see trapezoidal rule). The Euler-Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives f^(k) at the end points of the interval 0 and n. For any natural number p, we have The function f(x) (in blue) is approximated by a linear function (in red). ...

where B₁ = −1/2, B₂ = 1/6, B₃ = 0, B₄ = −1/30, B₅ = 0, B₆ = 1/42, B₇ = 0, B₈ = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

By employing the substitution rule, one can adapt this formula also to functions f which are defined on some other interval of the real line. In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

The remainder term

The remainder term R is given by

where are the periodic Bernoulli polynomials. The remainder term can be estimated as In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

Applications

If f is a polynomial and p is big enough, then the remainder term vanishes. For instance, if f(x) = x³, we can choose p = 2 to obtain after simplification In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

With the function f(x) = log(x), the Euler-Maclaurin formula can be used to derive precise error estimates for Stirling's approximation of the factorial function. The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ... The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a number n is the product of all positive integers less than or equal to n. ...

The Euler-Maclaurin formula is also used for detailed error analysis in numerical quadrature; in particular, extrapolation methods depend on it. In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. ...

Derivation

The Euler-MacLaurin formula can be understood as a curious application of some ideas from Hilbert spaces and functional analysis. Let B_n(x) be the Bernoulli polynomials. A set of functions dual to the Bernoulli polynomials are given by In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

where δ is the Dirac delta function. The above is a formal notation for the idea of taking derivatives at a point; thus one has The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

for n > 0 and some arbitrary but differentiable function f(x) on the unit interval. For the case of n = 0, one defines . The Bernoulli polynomials, along with their duals, form an orthogonal set of states on the unit interval: one has

and

The Euler-MacLaurin summation formula then follows as an integral over the latter. One has

Then taking x = 0, and rearranging terms, one obtains the traditional formula, together with the error term. Note that the Bernoulli numbers are defined as B_n = B_n(0), and that these vanish for odd n greater than 1. Note that this derivation does assume that f(x) is sufficiently differentiable and well-behaved; specifically, that f may be approximated by polynomials; equivalently, that f is a real analytic function. In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

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. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... In mathematics, the kernel of a function f may be taken to be either the equivalence relation on the functions domain that roughly expresses the idea of equivalent as far as the function f can tell, or the corresponding partition of the domain. ...

Motivation for the existence

From a formal point of view the existence of the Euler-MacLaurin summation formula can be motivated as follows. The difference operator Δ may formally be written as Δ = e^D − I, where D denotes an ordinary differential operator and I the identity operator (not the integral thus denoted above). Since the summation operator Σ is the inverse operator to the difference operator Δ we get In mathematics, a difference operator maps a function f(x) to another function f(x + a) − f(x + b). ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... Summation is the addition of a set of numbers; the result is their sum. ...

Now we know that the exponential generating function of the Bernoulli numbers is given by In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

hence formally

where denotes the integral operator. This purely formal derivation indicates the existence of the formula. The idea is due to Legendre. In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... Adrien-Marie Legendre (September 18, 1752 – January 10, 1833) was a French mathematician. ...

References

• Pierre Gaspard, "r-adic one-dimensional maps and the Euler summation formula", Journal of Physics A, 25 (letter) L483-L485 (1992). (Describes the eigenfunctions of the transfer operator for the Bernoulli map)
• Xavier Gourdon and Pascal Sebah, Introduction on Bernoulli's numbers, (2002)