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Leonard Euler was a Swiss mathematician, physicist, astronomer, logician, and engineer who made important and influential discoveries in many areas of mathematics. He also made pioneering contributions to topology and analytic number theory. Leonard Euler also introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the concept of mathematical function. He is also known for his work in mechanics, hydrodynamics, optics, astronomy, and music theory.

Euler is the only mathematician that has two named after him, the numbers: important Euler's number in calculus, e, approximately equal 2,71828, and the constant of Euler-Mascheroni γ (gamma), is approximately equal to 0,57721. It is unknown whether γ is rational or irrational.

First of all, Euler introduced the concept of the function and was the first to write f(x) to denote the function f applied to argument x. He also introduced modern notation for trigonometric functions, the letter E for the base of the natural logarithm (now also known as Euler number), the Greek letter Σ for summation, and the letter i for the imaginary unit. The use of the Greek letter π to denote the ratio of the circumference to its diameter was also popularized by Euler, although it originated with the Welsh mathematician William Jones.

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

In particular, Euler directly proved the power series expansion by e and the arctangent function. (An indirect proof using the inverse power series method was given by Newton and Leibniz between 1670 and 1680.) His bold use of the power series allowed him to solve the famous Basel problem in 1735 (he presented a more detailed argument in 1741)

Euler introduced the use of exponential function and logarithms in analytic proofs. He discovered ways to Express different logarithmic functions using power series and successfully defined logarithms for negative and complex numbers, thereby greatly expanding the field of mathematical applications of logarithms. He also defined the exponential function for complex numbers and discovered its relation with the trigonometric functions.

In addition, Euler developed the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He also invented the calculus of variations, including its most famous result is the equation of Euler-Lagrange.

Euler was also the first to use analytical methods to solve problems in number theory. At the same time he combined two disparate areas of mathematics and introduced a new area of research-analytical theory of numbers. In a breakthrough of this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions, and the analytic theory of continued fractions. For example, he proved the infinity of primes using the divergence of harmonic series, and he used analytical methods to get some idea of the way the primes are distributed. Euler's work in this field led to the development of the Prime number theorem.

Euler linked the nature of Prime distribution with ideas in analysis. He proved that the sum of inverse primes diverges. In doing so, he discovered a connection between the Riemann Zeta function and primes; this is known as the Euler formula for the Riemann Zeta function.

Euler proved Newton's identities, Fermat's small theorem, Fermat's theorem on sums of two squares, and he made a separate contribution to Lagrange's four squares theorem. He also invented the thotient function φ (n), a number of positive integers less than or equal to an integer n that are mutually simple to n. Using the properties of this function, he generalized Fermat's theorem to what is now known as Euler's theorem. He made a significant contribution to the theory of perfect numbers, which has fascinated mathematicians since Euclid. He proved that the relation shown between ideal numbers and Mersenne primes previously proved by Euclid was one-to-one, a result otherwise known as the Euclid-Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is considered as a fundamental theorem of number theory, and his ideas prepared the ground for the work of Karl Friedrich Gauss. By 1772, Euler proved that 231-1 = 2,147,483,647 is a simple Mersenne. It may have remained the largest known Prime Until 1867.

In 1735, Euler presented a solution to a problem known as the seven bridges of Koenigsberg. The city of Koenigsberg, Russia, was established on the Royal river and included two large Islands, which are connected by wires to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. This is impossible: there is no Eulerian scheme. This solution is considered to be the first theorem of graph theory, in particular the theory of planar graphs.

Euler also discovered a formula relating the number of vertices, edges, and faces of a convex polytope and hence a planar graph. The constant in this formula is now known as the Euler characteristic of a graph (or other mathematical object)and is related to the genus of the object. The study and generalization of this formula, in particular, Cauchy and L Hailie, is at the beginning of the topology

Some of Euler's greatest successes have been in solving real problems analytically, as well as in describing numerous applications of Bernoulli numbers, Fourier series, Euler numbers, constants e and π, continued fractions and integrals. He integrated Leibniz differential calculus with Newton's Fluxion method and developed tools that facilitated the application of calculus to physical problems. He made great strides in improving the numerical approximation of integrals by inventing what is now known as Eulerian approximations. The most notable of these approximations is the Euler and Euler–Maclaurin method formula. He also contributed to the use of differential equations, in particular the introduction of constants of Euler-Maceroni.

One of Euler's most unusual interests was the application of mathematical ideas to music. In 1739, he wrote the Tentamen Novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive much attention and was once described as too mathematical for musicians and too musical for mathematicians.

Euler helped develop the Euler-Bernoulli equation, which became the cornerstone of the technique. In addition to the successful application of its analytical tools to problems of classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy has been recognized by a number of Paris Academy awards during his career. His achievements include determining the orbits of comets and other celestial bodies with great accuracy, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.

In addition, Euler made an important contribution to optics. He disagreed with Newton's corpuscular theory of light in optics, which was then the prevailing theory. His 1740s work on optics helped ensure that the wave theory of light proposed by Christian Huygens would become the dominant way of thinking, at least until the development of quantum theory of light.

In 1757, he published an important set of equations for the inviscid flow, which are now known as the Euler equations.

Euler is also well known in construction engineering for its formula giving a critical buckling load to an ideal rack that depends only on its length and bending stiffness.

References

Александр Яковлев «Леонард Эйлер»

Елизавета Федоровна Литвинова, «Леонард Эйлер»

http://publ.lib.ru/ARCHIVES/E/EYLER_Leonard/_Eyler_L..html

https://ru.wikipedia.org/wiki/Эйлер,_Леонард

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