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.

* Leonhard Euler, ( 1707 – 1783 ), Swiss mathematician and physicist

* Carl Euler, ( 1834 – 1901 ), biologist

* Hans Heinrich Euler, ( 1901 – 1941 ), physicist

* Ulf von Euler, ( 1905 – 1983 ), Swedish physiologist, pharmacologist and Nobel laureate

* August Euler ( 1868 – 1957 ) German pioneer aviator

* William Daum Euler ( 1875 – 1961 ), Canadian politician

# REDIRECT Euler – Maclaurin formula

In mathematics, the Euler – Maclaurin formula provides a powerful connection between integrals ( see calculus ) and sums.

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

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

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.

and the error in this approximation is given by the Euler – Maclaurin formula:

where B < sub > k </ sub > is a Bernoulli number and R < sub > m, n </ sub > is the remainder term in the Euler – Maclaurin formula.

Because the remainder R < sub > m, n </ sub > in the Euler – Maclaurin formula satisfies

According to the Euler Maclaurin formula applied for

* 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.

For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic.

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.

The Euler – MacLaurin summation formula then follows as an integral over the latter.

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 ).

Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces and body forces.

Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture.

Euler proved in 1744 that the catenary is the curve which, when rotated about the x-axis, gives the surface of minimum surface area ( the catenoid ) for the given bounding circles.

If objects are seen as moving within a rotating frame, this movement results in another fictitious force, the Coriolis force ; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced.

When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces — the Coriolis force and the Euler force — will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the vector co-rotates with the planet.

If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the Euler force.

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

Another often used model, especially in computational fluid dynamics, is to use the Euler equations away from the body and the boundary layer equations, which incorporates viscosity, in a region close to the body.