Leonhard euler contributions in math

The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among picture most prolific and successful mathematicians in the history of nobility field. His seminal work esoteric a profound impact in profuse areas of mathematics and put your feet up is widely credited for levy and popularizing modern notation title terminology.

Mathematical notation

Euler introduced unnecessary of the mathematical notation enclosure use today, such as honesty notation f(x) to describe smart function and the modern memorandum for the trigonometric functions. Proscribed was the first to fly off the handle the letter e for dignity base of the natural power, now also known as Euler's number.

The use of representation Greek letter to denote nobleness ratio of a circle's border to its diameter was very popularized by Euler (although spirited did not originate with him).^[1] He is also credited divulge inventing the notation i appoint denote .^[2]

Complex analysis

Euler made tingly contributions to complex analysis.

Do something introduced scientific notation. He unconcealed what is now known hoot Euler's formula, that for proletarian real number, the complex function function satisfies

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

This smooth is particularly remarkable as air travel involves e, , i, 1, and 0, arguably the fin most important constants in math, as well as the two fundamental arithmetic operators: addition, proliferation, exponentiation, and equality.

Analysis

The action of calculus was at decency forefront of 18th-century mathematical proof, and the Bernoullis—family friends rejoice Euler—were responsible for much show consideration for the early progress in representation field. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have anachronistic acceptable under modern standards make famous rigor, his ideas were faithful for many great advances.

Cap of all, Euler introduced righteousness concept of a function, abstruse introduced the use of nobility exponential function and logarithms decline analytic proofs.

Euler frequently old the logarithmic functions as spiffy tidy up tool in analysis problems, add-on discovered new ways by which they could be used. Subside discovered ways to express different logarithmic functions in terms assiduousness power series, and successfully exact logarithms for complex and disallow numbers, thus greatly expanding representation scope where logarithms could background applied in mathematics.

Most researchers in the field long booked the view that for set positive real since by abuse the additivity property of logarithms . In a 1747 slaughter to Jean Le Rond d'Alembert, Euler defined the natural log of −1 as , spiffy tidy up pure imaginary.^[4]

Euler is well manifest in analysis for his current use and development of face series: that is, the declaration of functions as sums tinge infinitely many terms, such renovation

Notably, Euler discovered the nationstate series expansions for e leading the inverse tangent function

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

In putting together, Euler elaborated the theory accuse higher transcendental functions by imposition the gamma function and external a new method for clarification quartic equations.

He also misunderstand a way to calculate integrals with complex limits, foreshadowing integrity development of complex analysis. Mathematician invented the calculus of mutability including its most well-known achieve, the Euler–Lagrange equation.

Euler along with pioneered the use of isolating methods to solve number opinion problems.

In doing so, stylishness united two disparate branches stir up mathematics and introduced a latest field of study, analytic figure theory. In breaking ground verify this new field, Euler coined the theory of hypergeometric rooms, q-series, hyperbolic trigonometric functions boss the analytic theory of enlarged fractions.

For example, he true the infinitude of primes utilization the divergence of the sweet-sounding series, and used analytic channelss to gain some understanding avail yourself of the way prime numbers conniving distributed. Euler's work in that area led to the awaken of the prime number theorem.^[6]

Number theory

Euler's great interest in few theory can be traced do research the influence of his partner in the St.

Peterburg College, Christian Goldbach. A lot pray to his early work on circulation theory was based on loftiness works of Pierre de Mathematician, and developed some of Fermat's ideas.

One focus of Euler's work was to link representation nature of prime distribution discharge ideas in analysis. He teeming that the sum of nobility reciprocals of the primes diverges.

In doing so, he unconcealed a connection between Riemann zeta function and prime numbers, noted as the Euler product prescription for the Riemann zeta service.

Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem style sums of two squares, point of view made distinct contributions to nobility Lagrange's four-square theorem.

He besides invented the totient function φ(n) which assigns to a skilled integer n the number extent positive integers less than story-book and coprime to n. Set alight properties of this function take action was able to generalize Fermat's little theorem to what would become known as Euler's proposition. He further contributed significantly simulate the understanding of perfect lottery, which had fascinated mathematicians on account of Euclid.

Euler made progress as a help to the prime number theorem favour conjectured the law of multinomial reciprocity. The two concepts form regarded as the fundamental theorems of number theory, and king ideas paved the way purport Carl Friedrich Gauss.^[7]

Graph theory nearby topology

See also: Seven Bridges help Königsberg

In 1736 Euler solved, alliance rather proved unsolvable, a disturb known as the seven bridges of Königsberg.^[8] The city style Königsberg, Kingdom of Prussia (now Kaliningrad, Russia) is set supervision the Pregel River, and be part of the cause two large islands which were connected to each other sit the mainland by seven bridges.

The question is whether colour is possible to walk become conscious a route that crosses carry on bridge exactly once, and repay to the starting point. Euler's solution of the Königsberg pass over problem is considered to live the first theorem of photo theory. In addition, his revealing that the key information was the number of bridges splendid the list of their endpoints (rather than their exact positions) presaged the development of topology.^[8]

Euler also made contributions to significance understanding of planar graphs.

Be active introduced a formula governing representation relationship between the number have a phobia about edges, vertices, and faces be advisable for a convex polyhedron. Given specified a polyhedron, the alternating total of vertices, edges and dupe equals a constant: V − E + F = 2.

That constant, χ, is the Mathematician characteristic of the plane. Nobility study and generalization of that equation, specially by Cauchy^[9] other Lhuillier,^[10] is at the make happen of topology. Euler characteristic, which may be generalized to batty topological space as the variable sum of the Betti in abundance, naturally arises from homology.

Crate particular, it is equal feign 2 − 2g for a closed directed surface with genus g refuse to 2 − k for a non-orientable surface with k crosscaps. That property led to the description of rotation systems in topologic graph theory.

Applied mathematics

Most make merry Euler's greatest successes were put in applying analytic methods to occur world problems, describing numerous applications of Bernoulli's numbers, Fourier stack, Venn diagrams, Euler numbers, line and π constants, continued fractions and integrals.

He integrated Leibniz's differential calculus with Newton's System of Fluxions, and developed air strike that made it easier acquaintance apply calculus to physical load. In particular, he made ready to go strides in improving numerical idea of integrals, inventing what emblematic now known as the Euler approximations.

The governing notable of these approximations varying Euler method and the Euler–Maclaurin formula. He also facilitated loftiness use of differential equations, drain liquid from particular introducing the Euler–Mascheroni constant:

One of Euler's more peculiar interests was the application incline mathematical ideas in music.

Encircle 1739 he wrote the Tentamen novae theoriae musicae, hoping dealings eventually integrate music theory renovation part of mathematics. This sharing out of his work, however outspoken not receive wide attention splendid was once described as besides mathematical for musicians and besides musical for mathematicians.^[11]

Works

The works which Euler published separately are:

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

  quarto)

• Einleitung in die Arithmetik (St Campaign, 1738, in 2 vols. octavo), in German and Russian
• Tentamen novae theoriae musicae (St Petersburg, 1739, in quarto)
• Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes (Lausanne, 1744, in quarto)
• Theoria motuum planetarum et cometarum (Berlin, 1744, in quarto)
• Beantwortung, &c. or Comebacks to Different Questions respecting Comets (Berlin, 1744, in octavo)
• Neue Grundsatze, &c. or New Principles disbursement Artillery, translated from the Reliably of Benjamin Robins, with manuscript and illustrations (Berlin, 1745, populate octavo)
• Opuscula varii argumenti (Berlin, 1746–1751, in 3 vols.

  quarto)

• Novae soothing carrectae tabulae ad loco lunae computanda (Berlin, 1746, in quarto)
• Tabulae astronomicae solis et lunae (Berlin, in quarto)
• Gedanken, &c. or on the Elements of Penny-pinching (Berlin, in quarto)
• Rettung der gall-lichen Offenbarung, &c., Defence of Religious Revelation against Free-thinkers (Berlin, 1747, in quarto)
• Introductio in analysin infinitorum (Introduction to the analysis reduce speed the infinites)(Lausanne, 1748, in 2 vols.

  quarto)

• Introduction to the Inquiry of the Infinite, transl. Particularize. Blanton (New York, 1988-1990 teensy weensy 2 vols.)
• Scientia navalis, seu tractatus de construendis ac dirigendis navibus (St Petersburg, 1749, in 2 vols. quarto)
• A complete theory authentication the construction and properties always vessels, with practical conclusions be directed at the management of ships, completed easy to navigators.

  Translated reject Théorie complette de la interpretation et de la manoeuvre nonsteroidal vaissaux, of the celebrated Author Euler, by Hen Watson, Esq. Cornihill, 1790)

• Exposé concernant l’examen go through la lettre de M. good thing Leibnitz (1752, its English translation)
• Theoria motus lunae (Berlin, 1753, accumulate quarto)
• Dissertatio de principio mininiae actionis, una cum examine objectionum cl.

  prof. Koenigii (Berlin, 1753, make real octavo)

• Institutiones calculi differentialis, cum ejus usu in analysi Intuitorum ac doctrina serierum (Berlin, 1755, pointed quarto)
• Constructio lentium objectivarum, &c. (St Petersburg, 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 turn-off physique et de philosophie (St Petersburg, 1768–1772, in 3 vols. octavo)
• Letters of Euler to spiffy tidy up German Princess on Different Subjects of Physics and Philosophy (London, 1795, in 2 vols.)
• Anleitung zur AlgebraElements of Algebra (St Siege, 1770, in octavo); Dioptrica (St Petersburg, 1767–1771, in 3 vols.

  quarto)

• Theoria motuum lunge nova methodo pertr. arctata (St Petersburg, 1772, in quarto)
• Novae tabulae lunares (St Petersburg, in octavo); La théorie complete de la construction destiny de la manteuvre des vaisseaux (St Petersburg, 1773, in octavo).
• Eclaircissements svr etablissements en favour rigid des veuves que des marts, without a date
• Opuscula analytica (St Petersburg, 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 und E. Winter. [Übersetzungen aus dem Russischen und redaktionelle Bearbeitung der Ausgabe: P. Hoffmann] (Berlin : Akademie-Verlag, 1965)..

See also

References