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

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.

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

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 ).

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

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.

Paul Euler was a friend of the Bernoulli family — Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard.

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.

In classical field theory, one writes down a Lagrangian density,, involving a field, φ ( x, t ), and possibly its first derivatives (∂ φ /∂ t and ∇ φ ), and then applies a field-theoretic form of the Euler – Lagrange equation.

and then applies the Euler – Lagrange equation, one obtains the equation of motion

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

The Euler – Lagrange equations of motion for the functional E are then given in local coordinates by

So for instance if you have a sphere with a " dent ", then its total curvature is 4π ( the Euler characteristic of a sphere being 2 ), no matter how big or deep the dent.

The geodesic curvature of geodesics being zero, and the Euler characteristic of T being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle.

More generally, if the polyhedron has Euler characteristic ( where g is the genus, meaning " number of holes "), then the sum of the defect is

If ( d < sub > n </ sub >: A < sub > n </ sub > → A < sub > n-1 </ sub >) is a chain complex such that all but finitely many A < sub > n </ sub > are zero, and the others are finitely generated abelian groups ( or finite dimensional vector spaces ), then we can define the Euler characteristic

If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler – Lagrange equation.

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

* Normalization: If E is a line bundle, then where is the Euler class of the underlying real vector bundle.

The Euler – Lagrange equation, then, is given by

In the two-dimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program ( actually a suite of programs ) for airfoil design and analysis.

If is a fibration with fiber F, with the base B path-connected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property:

An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v, the indegree of v equals the outdegree of v. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour.

Substituting V and E into the Euler relation solved for F, one then obtains

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.

A quantitative form of Dirichlet's theorem states that if N ≥ 2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1 / n, where n = φ ( N ) is the Euler totient function.