The 18th-century Swiss mathematicianLeonhard Euler (1707–1783) is among the most prolific and successful mathematicians in thehistory of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.

Euler introduced much of the mathematical notation in use today, such as the notationf(x) to describe a function and the modern notation for thetrigonometric functions. He was the first to use the lettere for the base of thenatural logarithm, now also known asEuler's number. The use of the Greek letter${\displaystyle \pi }$ to denote theratio of a circle's circumference to its diameter was also popularized by Euler (although it did not originate with him).^[1] He is also credited for inventing the notationi to denote${\displaystyle \sqrt{-1}}$.^[2]

Euler made important contributions tocomplex analysis. He introduced scientific notation. He discovered what is now known asEuler's formula, that for anyreal number${\displaystyle \phi }$, the complexexponential function satisfies

${\displaystyle e^{i\phi }=\cos \phi +i\sin \phi .}$

Richard Feynman called this "the most remarkable formula in mathematics".^[3]Euler's identity is a special case of this:

${\displaystyle e^{i\pi }+1=0.}$

This identity is particularly remarkable as it involvese,${\displaystyle \pi }$,i, 1, and 0, arguably the five most important constants in mathematics, as well as the four fundamental arithmetic operators: addition, multiplication, exponentiation, and equality.

The development ofcalculus 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. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards ofrigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of afunction, and introduced the use of theexponential function andlogarithms in analytic proofs.

Euler frequently used the logarithmic functions as a tool in analysis problems, and discovered new ways by which they could be used. He discovered ways to express various logarithmic functions in terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly expanding the scope where logarithms could be applied in mathematics. Most researchers in the field long held the view that${\displaystyle \log (x)=\log (-x)}$ for any positive real${\displaystyle x}$ since by using the additivity property of logarithms${\displaystyle 2\log (-x)=\log ((-x{)}^2)=\log (x^2)=2\log (x)}$. In a 1747 letter toJean Le Rond d'Alembert, Euler defined the natural logarithm of −1 as${\displaystyle i\pi }$, apure imaginary.^[4]

Euler is well known in analysis for his frequent use and development ofpower series: that is, the expression of functions as sums of infinitely many terms, such as

${\displaystyle e=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}=\underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots +\frac{1}{n!}\right).}$

Notably, Euler discovered the power series expansions fore and theinverse tangent function

${\displaystyle \arctan z=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{z}^{2n+1}}{2n+1}.}$

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

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{n^2}\right)=\frac{{\pi }^2}{6}.}$

In addition, Euler elaborated the theory of higher transcendental functions by introducing thegamma function and introduced a new method for solvingquartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development ofcomplex analysis. Euler invented thecalculus of variations including its most well-known result, theEuler–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 ofhypergeometric series,q-series,hyperbolic trigonometric functions and the analytic theory ofcontinued fractions. For example, he proved theinfinitude of primes using the divergence of the harmonic series, and used analytic methods to gain some understanding of the wayprime numbers are distributed. Euler's work in this area led to the development of theprime number theorem.^[6]

Euler's great interest in number theory can be traced to the influence of his friend in the St. Peterburg Academy,Christian Goldbach. A lot of his early work on number theory was based on the works ofPierre de Fermat, and developed some of Fermat's ideas.

One focus of Euler's work was to link the nature of prime distribution with ideas in analysis. He proved thatthe sum of the reciprocals of the primes diverges. In doing so, he discovered a connection between Riemann zeta function and prime numbers, known as theEuler product formula for the Riemann zeta function.

Euler provedNewton's identities,Fermat's little theorem,Fermat's theorem on sums of two squares, and made distinct contributions to theLagrange's four-square theorem. He also invented thetotient function φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n. Using properties of this function he was able to generalize Fermat's little theorem to what would become known asEuler's theorem. He further contributed significantly to the understanding ofperfect numbers, which had fascinated mathematicians sinceEuclid. Euler made progress toward the prime number theorem and conjectured the law ofquadratic reciprocity. The two concepts are regarded as the fundamental theorems of number theory, and his ideas paved the way forCarl Friedrich Gauss.^[7]

In 1736 Euler solved, or rather proved unsolvable, a problem known as the seven bridges of Königsberg.^[8] The city ofKönigsberg,Kingdom of Prussia (now Kaliningrad, Russia) is set on thePregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point.Euler's solution of the Königsberg bridge problem is considered to be the first theorem ofgraph theory. In addition, his recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development oftopology.^[8]

Euler also made contributions to the understanding ofplanar graphs. He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the alternating sum of vertices, edges and faces equals a constant:V − E + F = 2. This constant, χ, is theEuler characteristic of the sphere. The study and generalization of this equation, specially byCauchy^[9] and Lhuillier,^[10] is at the origin oftopology. Euler characteristic, which may be generalized to anytopological space as the alternating sum of theBetti numbers, naturally arises fromhomology. In particular, it is equal to 2 − 2g for a closed orientedsurface with genusg and to 2 − k for a non-orientable surface with k crosscaps. This property led to the definition ofrotation systems intopological graph theory.

Most of Euler's greatest successes were in applying analytic methods to real world problems, describing numerous applications ofBernoulli's numbers,Fourier series,Venn diagrams,Euler numbers,e andπ constants, continued fractions and integrals. He integratedLeibniz'sdifferential calculus with Newton'sMethod of Fluxions, and developed tools that made it easier to apply calculus to physical problems. In particular, he made great strides in improvingnumerical approximation of integrals, inventing what are now known as theEuler approximations. The most notable of these approximations areEuler method and theEuler–Maclaurin formula. He also facilitated the use ofdifferential equations, in particular introducing theEuler–Mascheroni constant:

${\displaystyle \gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots +\frac{1}{n}-\ln (n)\right).}$

One of Euler's more unusual interests was the application of mathematical ideas inmusic. In 1739 he wrote theTentamen novae theoriae musicae, hoping to eventually integratemusic 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.^[11]

The works which Euler published separately are:

• Dissertatio physica de sono (Dissertation on the physics of sound) (Basel, 1727, in quarto)
• Mechanica, sive motus scientia analytice; exposita (St Petersburg, 1736, in 2 vols. quarto)
• Einleitung in die Arithmetik (St Petersburg, 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)
  • Additamentum II (English translation)
• Theoria motuum planetarum et cometarum (Berlin, 1744, in quarto)
• Beantwortung, &c. or Answers to Different Questions respecting Comets (Berlin, 1744, in octavo)
• Neue Grundsatze, &c. or New Principles of Artillery, translated from the English of Benjamin Robins, with notes and illustrations (Berlin, 1745, in octavo)
• Opuscula varii argumenti (Berlin, 1746–1751, in 3 vols. quarto)
• Novae et correctae tabulae ad loco lunae computanda (Berlin, 1746, in quarto)
• Tabulae astronomicae solis et lunae (Berlin, in quarto)
• Gedanken, &c. or Thoughts on the Elements of Bodies (Berlin, in quarto)
• Rettung der gall-lichen Offenbarung, &c., Defence of Divine Revelation against Free-thinkers (Berlin, 1747, in quarto)
• Introductio in analysin infinitorum (Introduction to the analysis of the infinites)(Lausanne, 1748, in 2 vols. quarto)
• Introduction to the Analysis of the Infinite, transl. J. Blanton (New York, 1988-1990 in 2 vols.)
• Scientia navalis, seu tractatus de construendis ac dirigendis navibus (St Petersburg, 1749, in 2 vols. quarto)
• A complete theory of the construction and properties of vessels, with practical conclusions for the management of ships, made easy to navigators. Translated from Théorie complette de la construction et de la manoeuvre des vaissaux, of the celebrated Leonard Euler, by Hen Watson, Esq. Cornihill, 1790)
• Exposé concernant l’examen de la lettre de M. de Leibnitz (1752, itsEnglish translation)
• Theoria motus lunae (Berlin, 1753, in quarto)
• Dissertatio de principio mininiae actionis, una cum examine objectionum cl. prof. 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 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 de physique et de philosophie (St Petersburg, 1768–1772, in 3 vols. octavo)
• Letters of Euler to a German Princess on Different Subjects of Physics and Philosophy (London, 1795, in 2 vols.)
• Anleitung zur AlgebraElements of Algebra (St Petersburg, 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 et de la manoeuvre des vaisseaux (St Petersburg, 1773, in octavo).
• Eclaircissements svr etablissements en favour taut des veuves que des marts, without a date
• Opuscula analytica (St Petersburg, 1783–1785, in 2 vols. quarto). SeeF. 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)..

• List of things named after Leonhard Euler

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