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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.
* 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.
The Euler Maclaurin formula is also used for detailed error analysis in numerical quadrature.
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 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.

Euler and summation
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.
* Euler summation
Euler was responsible for many of the notations in use today: the use of a, b, c for constants and x, y, z for unknowns, e for the base of the natural logarithm, sigma ( Σ ) for summation, i for the imaginary unit, and the functional notation f ( x ).
# Euler Maclaurin summation formula:
Further terms in this error estimate are given by the Euler Maclaurin summation formula.
Methods of generating such expansions include the Euler Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms.
This operator has the general form ( summation convention: sum over repeated indices — in this case over the three Euler angles ):
* Euler summation

Euler and formula
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 formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 ( and later generalized as Darboux's formula ).
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 can
The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force, the Coriolis force, and the centrifugal force, respectively.
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.
From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.
Euler also suggested that the complex logarithms can have infinitely many values.
The Euler equations can be integrated along a streamline to get Bernoulli's equation.
For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler Mascheroni constant ):
In large parts of the flow viscosity may be neglected ; such an inviscid flow can be described mathematically through the Euler equations, resulting from the Navier-Stokes equations when the viscosity is neglected.
Neither the Navier-Stokes equations nor the Euler equations lend themselves to exact analytic solutions ; usually engineers have to resort to numerical solutions to solve them, however Euler's equation can be solved by making further simplifying assumptions.
The motion of the top can be described by three Euler angles: the tilt angle between the symmetry axis of the top and the vertical ; the azimuth of the top about the vertical ; and the rotation angle of the top about its own axis.
It can be defined as a change in direction of the rotation axis in which the second Euler angle ( nutation ) is constant.
The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime.
#: Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles.
Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:
* The proof that every Haefliger structure on a manifold can be integrated to a foliation ( this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one ).
The Euler number E < sub > 2n </ sub > can be expressed as a sum over the even partitions of 2n,
Furthermore, the idea can be traced back to a paper by Leonhard Euler published in 1727, some 80 years before Thomas Young's 1807 paper.
Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus.
Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.
The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n / 2 handles.
Components can be calculated from the derivatives of the parameters defining the moving frame ( Euler angles or rotation matrices )
The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system.

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