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.

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 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 ).

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 provides general type-test and type-conversion operators.

The Euler – Lotka equation provides a means of identifying the intrinsic growth rate.

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.