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

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

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.

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:

This algebra is quotiented over by the ideal generated by the Euler – Lagrange equations.

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.

Writing coordinates ( t, x ) = ( x < sup > 0 </ sup >, x < sup > 1 </ sup >, x < sup > 2 </ sup >, x < sup > 3 </ sup >) = x < sup > μ </ sup >, this form of the Euler – Lagrange equation is

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

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.

Other attempts were made by Euler ( 1749 ), de Foncenex ( 1759 ), Lagrange ( 1772 ), and Laplace ( 1795 ).

These last four attempts assumed implicitly Girard's assertion ; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p ( z ).

In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler – Lagrange equations and Hamilton's equations.

In this instance, sometimes the term refers to the differential equations that the system satisfies ( e. g., Newton's second law or Euler – Lagrange equations ), and sometimes to the solutions to those equations.

Her submission included the celebrated discovery of what is now known as the " Kovalevsky top ", which was subsequently shown ( by Liouville ) to be the only other case of rigid body motion, beside the tops of Euler and Lagrange, that is " completely integrable ".

As a result of surface area minimization, a surface will assume the smoothest shape it can ( mathematical proof that " smooth " shapes minimize surface area relies on use of the Euler – Lagrange equation ).

This principle results in the Euler – Lagrange equations,

In other words, they satisfy the Euler – Lagrange equations

Notice that the Euler – Lagrange equations imply

Again using the Euler – Lagrange equations we get

Using the Euler – Lagrange field equations

Now, for any N, because of the Euler – Lagrange theorem, on shell ( and only on-shell ), we have

The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph Lagrange.

Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided an analytical solution to the problem.

These were developed intensively from the second half of the eighteenth century ( by, for example, D ' Alembert, Euler, and Lagrange ) until the 1930s.

More general equations of fluid flow-the Euler equations-were published by Leonhard Euler in 1757.

The Euler equations were extended to incorporate the effects of viscosity in the first half of the 1800s, resulting in the Navier-Stokes equations.

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

The standard equations of inviscid flow are the Euler equations.

Another often used model, especially in computational fluid dynamics, is to use the Euler equations away from the body and the boundary layer equations, which incorporates viscosity, in a region close to the body.

The Euler equations can be integrated along a streamline to get Bernoulli's equation.

Variation of the pressure around an airfoil as obtained by a solution of the Euler equations.

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.

) 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.

These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.

While Maxwell's equations are consistent within special and general relativity, there are some quantum mechanical situations in which Maxwell's equations are significantly inaccurate: including extremely strong fields ( see Euler – Heisenberg Lagrangian ) and extremely short distances ( see vacuum polarization ).

Brahmagupta ( 628 CE ) started the systematic study of indefinite quadratic equations — in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler.