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The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

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## Some Related Sentences

Euler and –

Thābit's

**formula**was rediscovered by Fermat ( 1601__–__1665 )**and**Descartes ( 1596__–__1650 ), to whom it is sometimes ascribed**,****and**extended by__Euler__( 1707__–__1783 ).**The**Bernoulli numbers appear

**in**

**the**Taylor series expansions

**of**

**the**tangent

**and**hyperbolic tangent functions

**,**

**in**formulas

**for**

**the**

**sum**

**of**powers

**of**

**the**first positive integers

**,**

**in**

**the**

__Euler__

__–__

**Maclaurin**

**formula**

**,**

**and**

**in**

**expressions**

**for**certain values

**of**

**the**Riemann zeta function

**.**

He also studied

**and**proved some theorems on perfect powers**,**such as**the**Goldbach__–____Euler__theorem**,****and**made several notable contributions to analysis**.****The**reason why

__Euler__

**and**some other authors relate

**the**Cauchy

__–__Riemann equations with analyticity is that a major theorem

**in**complex analysis says that holomorphic functions are analytic

**and**viceversa

**.**

In mathematics

**,****the**__Euler____–__**Maclaurin****formula****provides**a powerful connection**between**integrals ( see calculus )**and**sums**.**__Euler__computed this

**sum**to 20 decimal places with only a few

**terms**

**of**

**the**

__Euler__

__–__

**Maclaurin**

**formula**

**in**1735

**.**

Clenshaw

__–__Curtis quadrature 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 ).
In

**the**context**of**computing asymptotic expansions**of**sums**and**series**,**usually**the**most useful form**of****the**__Euler____–__**Maclaurin****formula**is

Euler and Maclaurin

**The**

**formula**was discovered independently by Leonhard

__Euler__

**and**Colin

__Maclaurin__around 1735 (

**and**later generalized as Darboux's

**formula**).

__Euler__needed it to compute slowly converging infinite series while

__Maclaurin__used it to calculate integrals

**.**

In this way

**we**get a proof**of****the**__Euler__**–**__Maclaurin__summation**formula**by mathematical induction**,****in**which**the**induction step relies on integration by parts**and**on**the**identities**for**periodic Bernoulli functions**.**
* June 8

**–**Leonhard__Euler__writes to James Stirling describing**the**__Euler__**–**__Maclaurin__**formula****,**providing a connection**between**integrals**and**calculus**.**
This was followed by a memoir on

**the**theory**of****the**tides**,**to which**,**conjointly with**the**memoirs by__Euler__**and**Colin__Maclaurin__**,**a prize was awarded by**the**French Academy: these three memoirs contain all that was done on this subject**between****the**publication**of**Isaac Newton's Philosophiae Naturalis Principia Mathematica**and****the**investigations**of**Pierre-Simon Laplace**.**
where B

**<**sub**>****k**</ sub**>**is a Bernoulli**number****and**R**<**sub**>****m****,****n**</ sub**>**is**the**remainder term**in****the**__Euler__**–**__Maclaurin__**formula****.**
* June 8-Leonhard

__Euler__writes to James Stirling describing**the**__Euler__**–**__Maclaurin__**formula****,**providing a connection**between**integrals**and**calculus**.**
where

**ƒ**is a smooth function**,**you could use**the**__Euler__**–**__Maclaurin__**formula**to convert**the**series into an**integral****,**plus some corrections involving**derivatives****of**S ( x ), then**for**large values**of**a you could use " stationary phase " method to calculate**the****integral****and**give an approximate evaluation**of****the****sum****.**

Euler and formula

Euler's

__formula__**,**named after Leonhard__Euler__**,**is a mathematical__formula__**in**complex analysis that establishes**the**deep relationship**between****the**trigonometric functions**and****the**complex exponential function**.**
It was

__Euler__( presumably around 1740 ) who turned his attention to**the**exponential function instead**of**logarithms**,****and**obtained**the**correct__formula__now named after him**.****The**

__Euler__

**–**MacLaurin

__formula__can be understood as a curious application

**of**some ideas from Banach spaces

**and**functional analysis

**.**

This way one can obtain

**expressions****for****ƒ**(**n**),**n**= 0**,**1**,**2**,**..., N**,****and**adding them up gives**the**__Euler__**–**MacLaurin__formula__**.****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

**.**

For closed ( orientable or non-orientable ) surfaces with positive genus

**,****the**maximum**number****p****of**colors needed depends on**the**surface's__Euler__characteristic χ according to**the**__formula__**The**only exception to

**the**

__formula__is

**the**Klein bottle

**,**which has

__Euler__characteristic 0 ( hence

**the**

__formula__gives

**p**= 7 )

**and**requires 6 colors

**,**as shown by P

**.**Franklin

**in**1934 ( Weisstein ).

Euler and provides

**The**Gauss-Bonnet theorem links

**the**total curvature

**of**a surface to its

__Euler__characteristic

**and**

__provides__an important link

**between**local geometric properties

**and**global topological properties

**.**

Hierholzer's 1873 paper

__provides__a different method**for**finding__Euler__cycles that is more efficient than Fleury's algorithm:__Euler__

**–**Bernoulli beam theory ( also known as engineer's beam theory or classical beam theory ) is a simplification

**of**

**the**linear theory

**of**elasticity which

__provides__a means

**of**calculating

**the**load-carrying

**and**deflection characteristics

**of**beams

**.**

Named after Leonhard

__Euler__**,**it is a prototypical example**of**a q-series**,**a modular form**,****and**__provides__**the**prototypical example**of**a relation**between**combinatorics**and**complex analysis**.**0.120 seconds.