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 young Ampère, however, soon resumed his Latin lessons, which enable him to master the works of Leonhard Euler and Daniel Bernoulli.

Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field.

After finishing his studies he went on long educational voyages from 1710 to 1724 through Europe, visiting other German states, England, Holland, Italy, and France, meeting with many famous mathematicians, such as Gottfried Leibniz, Leonhard Euler, and Nicholas I Bernoulli.

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

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.

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.

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.

Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician.

It is named for the Dutch-Swiss mathematician and scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others.

Some notable mathematicians include Archimedes of Syracuse, Leonhard Euler, Carl Gauss, Johann Bernoulli, Jacob Bernoulli, Aryabhata, Brahmagupta, Bhaskara II, Nilakantha Somayaji, Omar Khayyám, Muhammad ibn Mūsā al-Khwārizmī, Bernhard Riemann, Gottfried Leibniz, Andrey Kolmogorov, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, Alexander Grothendieck, David Hilbert, Alan Turing, von Neumann, Kurt Gödel, Joseph-Louis Lagrange, Georg Cantor, William Rowan Hamilton, Carl Jacobi, Évariste Galois, Nikolay Lobachevsky, Rene Descartes, Joseph Fourier, Pierre-Simon Laplace, Alonzo Church, Nikolay Bogolyubov and Pierre de Fermat.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d ' Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.

Together Bernoulli and Euler tried to discover more about the flow of fluids.

Later, Nikolaus Bernoulli made the same assertion concerning the polynomial x < sup > 4 </ sup > − 4x < sup > 3 </ sup > + 2x < sup > 2 </ sup > + 4x + 4, but he got a letter from Euler in 1742 in which he was told that his polynomial happened to be equal to

The Fourier series is named in honour of Joseph Fourier ( 1768 – 1830 ), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d ' Alembert, and Daniel Bernoulli.

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

In Tractatus de proportionibus ( 1328 ), Bradwardine extended the theory of proportions of Eudoxus of Cnidus to anticipate the concept of exponential growth, later developed by the Bernoulli and Euler, with compound interest as a special case.

It immediately occupied the attention of Jakob Bernoulli and the Marquis de l ' Hôpital, but Leonhard Euler first elaborated the subject.

In the eighteenth century, two of the innovators of mathematical physics were Swiss: Daniel Bernoulli ( for contributions to fluid dynamics, and vibrating strings ), and, more especially, Leonhard Euler, ( for his work in variational calculus, dynamics, fluid dynamics, and many other things ).

He calculated the Euler – Mascheroni constant, perhaps somewhat eccentrically, to 236 decimal places and evaluated the Bernoulli numbers up to the 62nd.

The solutions are based on linear isotropic infinitesimal elasticity and Euler – Bernoulli beam theory.

He worked with Euler on elasticity and the development of the Euler-Bernoulli beam equation.

* 1750: Euler – Bernoulli beam equation

date-Leonhard Euler and Daniel Bernoulli develop the Euler – Bernoulli beam equation.

The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.

Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.

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.

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

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.

In number theory, the Euler numbers are a sequence E < sub > n </ sub > of integers defined by the following Taylor series expansion:

* 1746 — Leonhard Euler develops the wave theory of light refraction and dispersion

Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s.

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

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

In number theory, an odd composite integer n is called an Euler – Jacobi pseudoprime to base a, if a and n are coprime, and

Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot.

Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial.

Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology.

This fact, with this proof, appears in the 1736 paper of Leonhard Euler on the Seven Bridges of Königsberg that first began the study of graph theory.

* 1707 – 1783: Leonhard Euler developed the theory of buckling of columns

In the language of stable homotopy theory, the Chern class, Stiefel-Whitney class, and Pontryagin class are stable, while the Euler class is unstable.

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

In the theory of modular forms it is typical to have Euler products with quadratic polynomials in the denominator here.

* ( Provides an introductory discussion of the Euler product in the context of classical number theory.

However, when the flow problem is put into a non-dimensional form, the viscous Navier – Stokes equations converge for increasing Reynolds numbers towards the inviscid Euler equations, suggesting that the flow should converge towards the inviscid solutions of potential flow theory – having the zero drag of the d ' Alembert paradox.

The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735.