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http://en.wikipedia.org/wiki/Leonhard_Euler

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. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics.

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number), the Greek letter ∑for summations and the letter i to denote the imaginary unit. 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. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. 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.

Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions.

In addition, Euler elaborated 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 best-known result, the Euler-Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for 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 infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.

Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Venn diagrams, Euler numbers, the constants e and π, continued fractions and integrals. He integrated Leibniz's differential calculus with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler-Maclaurin formula.

Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, 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 important contributions in optics. He disagreed with Newton's corpuscular theory of light in the Opticks, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christian Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.

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

Euler has an extensive bibliography. His best known books include:

Elements of Algebra. This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations.

Introductio in analysin infinitorum (1748). English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).

Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutionum calculi integralis (1768 _1770).

Lettres à une Princesse d'Allemagne (Letters to a German Princess) (1768 -1772). Available online (in French). English translation, with notes, and a life of Euler, available online from Google Books: Volume 1, Volume 2

Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). The Latin title translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.

A definitive collection of Euler's works, entitled Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences.

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