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 characteristic polynomial of the matrix is given by the Euler – Lotka equation.

The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force, the Coriolis force, and the centrifugal force, respectively.

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

His correspondence with Euler ( who also knew the above equation ) shows that he didn't fully understand logarithms.

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

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.

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.

Euler was aware of the work of Lord Brouncker, the first European mathematician to find a general solution of the equation, but apparently confused Brouncker with Pell.

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

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions.

Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:

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

The Euler – Tricomi equation is used in the investigation of transonic flow.

Others such as the Euler – Tricomi equation have different types in different regions.

* 1739-Leonhard Euler solves the ordinary differential equation for a forced harmonic oscillator and notices the resonance phenomenon

* 1759-Leonhard Euler solves the partial differential equation for the vibration of a rectangular drum

* 1764-Leonhard Euler examines the partial differential equation for the vibration of a circular drum and finds one of the Bessel function solutions

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

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 is simply the Euler method for integrating the differential equation: