In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Euler–Maclaurin formula is a formula for the difference between an
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
and a closely related
sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
using integrals and the machinery of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. For example, many asymptotic expansions are derived from the formula, and
Faulhaber's formula
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers
:\sum_^n k^p = 1^p + 2^p + 3^p + \cdots + n^p
as a (''p''&nb ...
for the sum of powers is an immediate consequence.
The formula was discovered independently by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to
Darboux's formula.
The formula
If and are
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s and is a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
valued
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
for
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s in the
interval , then the integral
can be approximated by the sum (or vice versa)
(see
rectangle method
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s evaluated at the endpoints of the interval, that is to say and .
Explicitly, for a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
and a function that is times
continuously differentiable on the interval , we have
where is the th
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
(with ) and is an
error term In mathematics and statistics, an error term is an additive type of error. Common examples include:
* errors and residuals in statistics, e.g. in linear regression
* the error term in numerical integration
In analysis, numerical integration ...
which depends on , , , and and is usually small for suitable values of .
The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for . In this case we have
or alternatively
The remainder term
The remainder term arises because the integral is usually not exactly equal to the sum. The formula may be derived by applying repeated
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
to successive intervals for . The boundary terms in these integrations lead to the main terms of the formula, and the leftover integrals form the remainder term.
The remainder term has an exact expression in terms of the periodized Bernoulli functions . The Bernoulli polynomials may be defined recursively by and, for ,
The periodized Bernoulli functions are defined as
where denotes the largest integer less than or equal to , so that always lies in the interval .
With this notation, the remainder term equals
When , it can be shown that
where denotes the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
; one approach to prove this inequality is to obtain the Fourier series for the polynomials . The bound is achieved for even when is zero. The term may be omitted for odd but the proof in this case is more complex (see Lehmer).
Using this inequality, the size of the remainder term can be estimated as
Low-order cases
The Bernoulli numbers from to are . Therefore the low-order cases of the Euler–Maclaurin formula are:
Applications
The Basel problem
The
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
is to determine the sum
Euler computed this sum to 20 decimal places with only a few terms of the Euler–Maclaurin formula in 1735. This probably convinced him that the sum equals , which he proved in the same year.
Sums involving a polynomial
If is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
and is big enough, then the remainder term vanishes. For instance, if , we can choose to obtain, after simplification,
Approximation of integrals
The formula provides a means of approximating a finite integral. Let be the endpoints of the interval of integration. Fix , the number of points to use in the approximation, and denote the corresponding step size by . Set , so that and . Then:
This may be viewed as an extension of the
trapezoid rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral.
\int_a^b f(x) \, dx.
The trapezoidal rule works by ...
by the inclusion of correction terms. Note that this asymptotic expansion is usually not convergent; there is some , depending upon and , such that the terms past order increase rapidly. Thus, the remainder term generally demands close attention.
The Euler–Maclaurin formula is also used for detailed
error analysis in
numerical quadrature
In analysis, numerical integration comprises 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 ...
. It explains the superior performance of the
trapezoidal rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral.
\int_a^b f(x) \, dx.
The trapezoidal rule works b ...
on smooth
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
s and is used in certain
extrapolation methods.
Clenshaw–Curtis quadrature
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = \cos ...
is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler–Maclaurin approach is very accurate (in that particular case the Euler–Maclaurin formula takes the form of a
discrete cosine transform). This technique is known as a periodizing transformation.
Asymptotic expansion of sums
In the context of computing
asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
s of sums and
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
, usually the most useful form of the Euler–Maclaurin formula is
where and are integers.
Often the expansion remains valid even after taking the limits or or both. In many cases the integral on the right-hand side can be evaluated in
closed form in terms of
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
s even though the sum on the left-hand side cannot. Then all the terms in the asymptotic series can be expressed in terms of elementary functions. For example,
Here the left-hand side is equal to , namely the first-order
polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
defined by
:
the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
is equal to when is a
positive integer. This results in an asymptotic expansion for . That expansion, in turn, serves as the starting point for one of the derivations of precise error estimates for
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
of the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
function.
Examples
If is an integer greater than 1 we have:
Collecting the constants into a value of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, we can write an asymptotic expansion:
For equal to 2 this simplifies to
or
When , the corresponding technique gives an asymptotic expansion for the
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
s:
where is the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
.
Proofs
Derivation by mathematical induction
We outline the argument given in Apostol.
The
Bernoulli polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.
These polynomials occur in ...
and the periodic Bernoulli functions for were introduced above.
The first several Bernoulli polynomials are
The values are the
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
. Notice that for we have
and for ,
The functions agree with the Bernoulli polynomials on the interval and are
periodic with period 1. Furthermore, except when , they are also continuous. Thus,
Let be an integer, and consider the integral
where
Integrating by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
, we get
Using , , and summing the above from to , we get
Adding to both sides and rearranging, we have
This is the case of the summation formula. To continue the induction, we apply integration by parts to the error term:
where
The result of integrating by parts is
Summing from to and substituting this for the lower order error term results in the case of the formula,
This process can be iterated. In this way we get a proof of the Euler–Maclaurin summation formula which can be formalized by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
, in which the induction step relies on integration by parts and on identities for periodic Bernoulli functions.
See also
*
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
*
Euler summation
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ'' ...
*
Gauss–Kronrod quadrature formula
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information ...
*
Darboux's formula
*
Euler–Boole summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by
: \frac=\sum_^\infty E_n(x)\frac.
The concept is named after Leonhard Euler and George Boole.
The periodic Euler functions are ...
References
Further reading
*
*
*
*
External links
*
{{Calculus topics
Approximation theory
Asymptotic analysis
Hilbert space
Numerical integration (quadrature)
Articles containing proofs
Theorems in analysis
Summability methods
Leonhard Euler