Leonhard Euler

Leonhard_Euler-540x262

In mathematics circles many think of Leonhard Euler as the most preeminent mathematician of the 18th century, and one of the greatest mathematicians to have ever lived.  The former Soviet Union celebrated the life and legacy of Leonhard Euler on a 1957 Stamp with an image and the words  “250 years from the birth of the great mathematician, academician Leonhard Euler.”

Born on the 15th of April 1707 in Basel Switzerland, Euler was a mathematician and physicist, and his greatest legacy was his involvement in modern day mathematics notation and terminology. He also made important discoveries in fields as diverse as infinitesimal calculus and graph theory.

Euler is the only mathematician to have two numbers named after him: the immensely important Euler’s Number in calculus, e, approximately equal to 2.71828, and the Euler-Mascheroni Constant γ (gamma) sometimes referred to as just “Euler’s constant”, approximately equal to 0.57721. It is not known whether γ is rational or irrational. In 2002 Euler was further honored by having the asteroid Euler named after him.

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 died on the 18th of September 1783 in Saint Petersburg Russia.

Leonhard Euler

Today Google Doodle honors the Scientist Leonhard Euler. Also in today’s Google Doodle one can see the Mathematical Constant e graphic, which formed part of Leonhard Euler’s mathematic theories and calculations.

Mathematical notation

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 Σ 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.

Analysis

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[25] (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, such as

e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):

\sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.

A geometric interpretation of Euler’s formula

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.[23] He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler’s formula states that the complex exponential function satisfies

e^{i\varphi} = \cos \varphi + i\sin \varphi.\,

A special case of the above formula is known as Euler’s identity,

e^{i \pi} +1 = 0 \,

called “the most remarkable formula in mathematics” by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, ei and π. In 1988, readers of the Mathematical Intelligencer voted it “the Most Beautiful Mathematical Formula Ever”. In total, Euler was responsible for three of the top five formulae in that poll.

De Moivre’s formula is a direct consequence of Euler’s formula.

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.

Leonhard-Euler-german note

In Germany they have also honored Leonhard Euler on a stamp commemorating on the 200th anniversary of his death. Across the center of the stamp it shows his polyhedral formula, nowadays written as v − e + f = 2. This formula is part of the Google Doodle today celebrating the 306th year since the birth of Leonhard Euler.

Number theory

Euler’s interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler’s early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat’s ideas, and disproved some of his conjectures.

Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function.

Euler proved Newton’s identities, Fermat’s little theorem, Fermat’s theorem on sums of two squares, and he made distinct contributions to Lagrange’s four-square theorem. He also invented the totient function φ(n) which is the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat’s little theorem to what is now known as Euler’s theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss.

By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.

Graph theory

In 1736, Euler solved the problem known as the Seven Bridges of Königsberg. The city of Königsberg, Prussia was set on the Pregel River, and included two large islands which were connected 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. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem ofgraph theory, specifically of planar graph theory.

Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convexpolyhedron, and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy[34] and L’Huillier, is at the origin of topology.

Applied mathematics

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 approximationof integrals, inventing what are now known as the Euler approximations. The most notable of these approximations areEuler’s method and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant:

\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).

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

Leonhard_Euler_sixrthseries

Euler was also featured on the Switzerland sixth Series of the 10-franc note. On the note the Swizz printed a fine-art image of Leonhard Euler and the note also had his name on it and the date of his birth and death.

Physics and astronomy

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.

Logic

Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.

Leonhard_Euler

A 1753 portrait by Emanuel Handmann. This portrayal suggests problems of the right eyelid, and possible strabismus. The left eye, which here appears healthy, was later affected by a cataract.

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