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 equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

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 ).

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

This view was further elaborated by Belidor ( representation of rough surfaces with spherical asperities, 1737 ) and Leonhard Euler ( 1750 ) who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction ..

Euler was born on April 15, 1707, in Basel to Paul Euler, a pastor of the Reformed Church.

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.

As a result, it was made especially attractive to foreign scholars like Euler.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731.

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece.

This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy.

Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes.

For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last.

The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.

This direct relationship between curved streamlines and pressure differences was derived from Newton's second law by Leonard Euler in 1754:

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

Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

Leonhard Euler, for whom the concept is named, was responsible for much of this early work.

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.

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.

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.

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

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

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.

This way one can obtain expressions for ƒ ( n ), n = 0, 1, 2, ..., N, and adding them up gives the Euler – MacLaurin formula.

Sometimes is used, which is unfortunate since this is also used for the Euler characteristic

In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy.

Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for and the inverse tangent function.

Euler diagram for P ( complexity ) | P, NP, NP-complete, and NP-hard set of problems.

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.

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.

Euler also suggested that the complex logarithms can have infinitely many values.

) A vortex flow of any strength may be added to this uniform flow and the equation is solved, thus there are many flows that solve the Euler equations.

The concept was introduced by Leonhard Euler in his 1765 book Theoria motus corporum solidorum seu rigidorum ; he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.

Under this heading, the Board made many lesser awards, including some awards in total £ 5, 000 made to John Harrison before he received his main prize, an award of £ 3, 000 to the widow of Tobias Mayer, whose lunar tables were the basis of the lunar data in the early decades of the Nautical Almanac, £ 300 to Leonhard Euler for his ( assumed ) contribution to the work of Mayer, £ 50 each to Richard Dunthorne and Israel Lyons for contributing methods to shorten the calculations connected with lunar distances, and awards made to the designers of improvements in chronometers.

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

In modern mathematics, the Euler characteristic arises from homology and connects to many other invariants.

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 ).

It can be shown that for any odd composite n, at least ¾ of the bases a are witnesses for the compositeness of n. The Miller – Rabin test is strictly stronger than the Solovay – Strassen primality test in the sense that for every composite n, the set of strong liars for n is a subset of the set of Euler liars for n, and for many n, the subset is proper.

Impellers are designed in many configurations, and Euler ’ s pump and turbine equation plays an important role in understanding impeller performance.

However, many scholars, including Leonhard Euler, believe it originates from the letter " r ", the first letter of the Latin word " radix " ( meaning " root "), referring to the same mathematical operation.

Graham, prolific mathematician and industrious human being, has won many other prizes over the years ; he was one of the laureates of the prestigious Pólya Prize the first year it was ever awarded, and among the first to win the Euler Medal.

This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.

Later, it became a significant topic for Euler, who gave an explicit formula for all triangular numbers that are also perfect squares, among many other discoveries relating to figurate numbers.

When at home, Stone relied upon the research facilities and expertise made available to him by Esther Euler, head research librarian of the University of California at Los Angeles, to whom he dedicated and thanked, in addition to many others, in several of his works.

Although Le Sage published not many papers in his life, he had an extensive letter exchange to people like Jean le Rond d ' Alembert, Leonhard Euler, Paolo Frisi, Roger Joseph Boscovich, Johann Heinrich Lambert, Pierre Simon Laplace, Daniel Bernoulli, Firmin Abauzit, Lord Stanhope etc ..

Old Swiss 10 Franc Banknote honouring Leonhard Euler who developed many key concepts in mathematics, calculus, physics and engineering.

The description of the Hasse – Weil zeta function up to finitely many factors of its Euler product is relatively simple.

In 2009, AMS released version 3. 0 of AMS fonts, in which Hermann Zapf reshaped many of the Euler glyphs, with implementation and assistance from Hans Hagen, Taco Hoekwater, and Volker RW Schaa.

As soon as the punctured Euler characteristic is negative, there are only finitely many holomorphic reparametrizations of that preserve the marked points.

Typically one admits only those and that make the punctured Euler characteristic of negative ; then the domain is stable, meaning that there are only finitely many holomorphic automorphisms of that preserve the marked points.