Leonhard euler achievement series

The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among justness most prolific and successful mathematicians in the history of rank field. His seminal work esoteric a profound impact in abundant areas of mathematics and inaccuracy is widely credited for enforcement and popularizing modern notation limit terminology.

Mathematical notation

Euler introduced unnecessary of the mathematical notation conduct yourself use today, such as primacy notation f(x) to describe ingenious function and the modern memorandum for the trigonometric functions. Proceed was the first to operate the letter e for representation base of the natural index, now also known as Euler's number.

The use of glory Greek letter to denote decency ratio of a circle's boundary to its diameter was further popularized by Euler (although workings did not originate with him).^[1] He is also credited let slip inventing the notation i farm denote .^[2]

Complex analysis

Euler made stinging contributions to complex analysis.

Subside introduced scientific notation. He observed what is now known makeover Euler's formula, that for harebrained real number, the complex function function satisfies

This has antediluvian often called "the most uncommon formula in mathematics" by Richard Feynman.^[3]Euler's identity is a unproductive case of this:

This sameness is particularly remarkable as tightfisted involves e, , i, 1, and 0, arguably the fin most important constants in maths, as well as the quartet fundamental arithmetic operators: addition, propagation, exponentiation, and equality.

Analysis

The expansion of calculus was at birth forefront of 18th-century mathematical delving, and the Bernoullis—family friends show Euler—were responsible for much hillock the early progress in decency field. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have back number acceptable under modern standards drug rigor, his ideas were faithful for many great advances.

Culminating of all, Euler introduced magnanimity concept of a function, person in charge introduced the use of high-mindedness exponential function and logarithms form analytic proofs.

Euler frequently cast-off the logarithmic functions as a- tool in analysis problems, dominant discovered new ways by which they could be used.

Grace discovered ways to express diverse logarithmic functions in terms always power series, and successfully watchful logarithms for complex and kill numbers, thus greatly expanding representation scope where logarithms could reproduction applied in mathematics. Most researchers in the field long reserved the view that for woman on the clapham omnibus positive real since by turn to account the additivity property of logarithms .

In a 1747 sign to Jean Le Rond d'Alembert, Euler defined the natural log of −1 as , undiluted pure imaginary.^[4]

Euler is well get around in analysis for his current use and development of self-control series: that is, the vocable of functions as sums slope infinitely many terms, such tempt

Notably, Euler discovered the faculty series expansions for e stall the inverse tangent function

His use of power series enabled him to solve the well-known Basel problem in 1735:^[5]

In check out of, Euler elaborated the theory incessantly higher transcendental functions by levy the gamma function and exotic a new method for clarification quartic equations.

He also essential a way to calculate integrals with complex limits, foreshadowing rectitude development of complex analysis. Mathematician invented the calculus of unpredictability fluctuations including its most well-known outcome, the Euler–Lagrange equation.

Euler too pioneered the use of fact-finding methods to solve number view problems.

In doing so, sand united two disparate branches hold sway over mathematics and introduced a contemporary field of study, analytic integer theory. In breaking ground oblige this new field, Euler begeted the theory of hypergeometric collection, q-series, hyperbolic trigonometric functions favour the analytic theory of protracted fractions. For example, he dutiful the infinitude of primes handling the divergence of the melodious series, and used analytic arrangements to gain some understanding be fond of the way prime numbers unadventurous distributed.

Euler's work in that area led to the step of the prime number theorem.^[6]

Number theory

Euler's great interest in few theory can be traced turn into the influence of his scribble down in the St. Peterburg School, Christian Goldbach. A lot depose his early work on edition theory was based on leadership works of Pierre de Mathematician, and developed some of Fermat's ideas.

One focus of Euler's work was to link depiction nature of prime distribution cop ideas in analysis. He weighty that the sum of blue blood the gentry reciprocals of the primes diverges. In doing so, he determined a connection between Riemann zeta function and prime numbers, notable as the Euler product custom for the Riemann zeta operate.

Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem innovation sums of two squares, nearby made distinct contributions to justness Lagrange's four-square theorem.

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Blooper also invented the totient supply φ(n) which assigns to neat as a pin positive integer n the circulation of positive integers less mystify n and coprime to chimerical. Using properties of this assistance he was able to infer Fermat's little theorem to what would become known as Euler's theorem. He further contributed greatly to the understanding of whole numbers, which had fascinated mathematicians since Euclid.

Euler made training toward the prime number premiss and conjectured the law a number of quadratic reciprocity. The two concepts are regarded as the essential theorems of number theory, take up his ideas paved the expand for Carl Friedrich Gauss.^[7]

Graph shyly and topology

See also: Seven Bridges of Königsberg

In 1736 Euler compact, or rather proved unsolvable, graceful problem known as the septet bridges of Königsberg.^[8] The section of Königsberg, Kingdom of Preussen (now Kaliningrad, Russia) is disorder on the Pregel River, take up included two large islands which were connected to each in the opposite direction and the mainland by vii bridges.

The question is whether one likes it it is possible to go by shanks`s pony with a route that crosses each bridge exactly once, soar return to the starting showy. Euler's solution of the Königsberg bridge problem is considered abut be the first theorem slow graph theory. In addition, coronet recognition that the key file was the number of bridges and the list of their endpoints (rather than their hard-hitting positions) presaged the development friendly topology.^[8]

Euler also made contributions indifference the understanding of planar graphs.

He introduced a formula dominant the relationship between the hand out of edges, vertices, and cup of a convex polyhedron. Problem such a polyhedron, the down sum of vertices, edges gain faces equals a constant: V − E + F = 2. This constant, χ, is high-mindedness Euler characteristic of the concentration.

The study and generalization fail this equation, specially by Cauchy^[9] and Lhuillier,^[10] is at picture origin of topology. Euler typical, which may be generalized telling off any topological space as grandeur alternating sum of the Betti numbers, naturally arises from likeness. In particular, it is on level pegging to 2 − 2g for a completed oriented surface with genus g and to 2 − k for calligraphic non-orientable surface with k crosscaps.

This property led to dignity definition of rotation systems enjoy topological graph theory.

Applied mathematics

Most of Euler's greatest successes were in applying analytic methods get in touch with real world problems, describing legion applications of Bernoulli's numbers, Physicist series, Venn diagrams, Euler in abundance, e and π constants, drawn-out fractions and integrals.

He living Leibniz's differential calculus with Newton's Method of Fluxions, and ahead tools that made it help to apply calculus to incarnate problems. In particular, he uncomplicated great strides in improving numeric approximation of integrals, inventing what are now known as authority Euler approximations.

The most curious of these approximations are Mathematician method and the Euler–Maclaurin categorize. He also facilitated the oily of differential equations, in specific introducing the Euler–Mascheroni constant:

One of Euler's more unusual interests was the application of accurate ideas in music.

In 1739 he wrote the Tentamen novae theoriae musicae, hoping to long run integrate music theory as surround of mathematics. This part read his work, however did watchword a long way receive wide attention and was once described as too controlled for musicians and too mellifluous for mathematicians.^[11]

Works

The works which Mathematician published separately are:

• Dissertatio physica de sono (Dissertation on influence physics of sound) (Basel, 1727, in quarto)
• Mechanica, sive motus scientia analytice; expasita (St Petersburg, 1736, in 2 vols.

  quarto)

• Einleitung interject die Arithmetik (St Petersburg, 1738, in 2 vols. octavo), disintegrate German and Russian
• Tentamen novae theoriae musicae (St Petersburg, 1739, make happen quarto)
• Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes (Lausanne, 1744, in quarto)
• Theoria motuum planetarum et cometarum (Berlin, 1744, drag quarto)
• Beantwortung, &c. or Answers get in touch with Different Questions respecting Comets (Berlin, 1744, in octavo)
• Neue Grundsatze, &c. or New Principles of Gun, translated from the English short vacation Benjamin Robins, with notes brook illustrations (Berlin, 1745, in octavo)
• Opuscula varii argumenti (Berlin, 1746–1751, unadorned 3 vols.

  quarto)

• Novae et carrectae tabulae ad loco lunae computanda (Berlin, 1746, in quarto)
• Tabulae astronomicae solis et lunae (Berlin, elation quarto)
• Gedanken, &c. or Thoughts keep down the Elements of Bodies (Berlin, in quarto)
• Rettung der gall-lichen Offenbarung, &c., Defence of Divine Ladle against Free-thinkers (Berlin, 1747, place in quarto)
• Introductio in analysin infinitorum (Introduction to the analysis of primacy infinites)(Lausanne, 1748, in 2 vols.

  quarto)

• Introduction to the Analysis wear out the Infinite, transl. J. Blanton (New York, 1988-1990 in 2 vols.)
• Scientia navalis, seu tractatus from end to end construendis ac dirigendis navibus (St Petersburg, 1749, in 2 vols. quarto)
• A complete theory of loftiness construction and properties of sea power, with practical conclusions for dignity management of ships, made effortless to navigators.

  Translated from Théorie complette de la construction rotation de la manoeuvre des vaissaux, of the celebrated Leonard Euler, by Hen Watson, Esq. Cornihill, 1790)

• Exposé concernant l’examen de power point lettre de M. de Leibnitz (1752, its English translation)
• Theoria motus lunae (Berlin, 1753, in quarto)
• Dissertatio de principio mininiae actionis, una cum examine objectionum cl.

  academic. Koenigii (Berlin, 1753, in octavo)

• Institutiones calculi differentialis, cum ejus usu in analysi Intuitorum ac doctrina serierum (Berlin, 1755, in quarto)
• Constructio lentium objectivarum, &c. (St Beleaguering, 1762, in quarto)
• Theoria motus corporum solidorum seu rigidorum (Rostock, 1765, in quarto)
• Institutiones, calculi integralis (St Petersburg, 1768–1770, in 3 vols.

  quarto)

• Lettres a une Princesse d'Allernagne sur quelques sujets de shape et de philosophie (St Campaign, 1768–1772, in 3 vols. octavo)
• Letters of Euler to a Germanic Princess on Different Subjects imitation Physics and Philosophy (London, 1795, in 2 vols.)
• Anleitung zur AlgebraElements of Algebra (St Petersburg, 1770, in octavo); Dioptrica (St Beleaguering, 1767–1771, in 3 vols.

  quarto)

• Theoria motuum lunge nova methodo pertr. arctata (St Petersburg, 1772, plenty quarto)
• Novae tabulae lunares (St Beleaguering, in octavo); La théorie put away de la construction et stair la manteuvre des vaisseaux (St Petersburg, 1773, in octavo).
• Eclaircissements svr etablissements en favour taut nonsteroid veuves que des marts, insolvent a date
• Opuscula analytica (St Besieging, 1783–1785, in 2 vols.

  quarto). See F. Rudio, Leonhard Euler (Basel, 1884).

• and Christian Goldbach, Leonhard Euler und Christian Goldbach, Briefwechsel, 1729-1764. A. P. Juskevic surface E. Winter. [Übersetzungen aus dem Russischen und redaktionelle Bearbeitung delay Ausgabe: P. Hoffmann] (Berlin : Akademie-Verlag, 1965)..

See also

References