List of things named after Leonhard Euler

Leonhard Euler (1707–1783)

In mathematics and physics, there are a large number of topics named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.

Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.^[1]^[2]

Euler's conjectures

Euler's equations

Usually, Euler's equation refers to one of (or a set of) differential equations (DEs). It is customary to classify them into ODEs and PDEs.

Otherwise, Euler's equation might refer to a non-differential equation, as in these three cases:

Euler–Lotka equation, a characteristic equation employed in mathematical demography
Euler's pump and turbine equation
Euler transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric series

Euler's ordinary differential equations

Euler rotation equations, a set of first-order ODEs concerning the rotations of a rigid body.
Euler–Cauchy equation, a linear equidimensional second-order ODEs with variable coefficients. Its second-order version can emerge from Laplace equation in polar coordinates.
Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams.
Euler–Lagrange equation, a second-order ODE emerging from minimization problems in calculus of variations.

Euler's partial differential equations

Euler conservation equations, a set of quasilinear first-order hyperbolic equations used in fluid dynamics for inviscid flows. In the (Froude) limit of no external field, they are conservation equations.
Euler–Tricomi equation – a second-order PDE emerging from Euler conservation equations.
Euler–Poisson–Darboux equation, a second-order PDE playing important role in solving the wave equation.

Euler's formulas

Euler's formula in complex analysis $e ix = cos x + i sin x$
Euler's polyhedral formula for planar graphs or polyhedra: $v - e + f = 2$
Euler's formula for the critical load of a column: $P_\text{cr}=\frac{\pi^2 EI}{(KL)^2}$
Euler's continued fraction formula connecting a finite sum of products with a finite continued fraction
Euler product formula for the Riemann zeta function.
Euler–Maclaurin formula (Euler's summation formula) relating integrals to sums
Euler–Rodrigues formula describing the rotation of a vector in three dimensions

Euler's functions

The Euler function, a modular form that is a prototypical q-series.
Euler's homogeneous function theorem
Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.
Euler hypergeometric integral

Euler's identities

Euler's identity $e i π + 1 = 0$ .
Euler's four-square identity, which shows that the product of two sums of four squares can itself be expressed as the sum of four squares.
Euler's identity may also refer to the pentagonal number theorem.

Euler's numbers

Euler's number, $e \approx 2.71828$ , the base of the natural logarithm, also known as Napier's constant.
Euler's idoneal numbers, a set of 65 or possibly 66 integers with special properties
Euler numbers are an integer sequence defined by the Taylor series expansion of the hyperbolic secant.
Eulerian numbers count certain types of permutations.
Euler number (physics), the cavitation number in fluid dynamics.
Euler number (topology) – now, Euler characteristic, classically the number of vertices minus edges plus faces of a polyhedron.
Lucky numbers of Euler
Euler–Mascheroni constant, $γ \approx 0.5772$ , the limiting difference between the harmonic series and the natural logarithm.
Eulerian integers are the numbers of form $a + bω$ where $ω$ is a complex cube root of 1.

Euler's theorems

Euler's homogeneous function theorem, a theorem about homogeneous polynomials.
Euler's infinite tetration theorem
Euler's rotation theorem
Euler's theorem (differential geometry) on the existence of the principal curvatures of a surface and orthogonality of the associated principal directions.
Euler's theorem in geometry, relating the circumcircle and incircle of a triangle.
Euler's quadrilateral theorem, an extension of the parallelogram law to convex quadrilaterals
Euclid–Euler theorem, relating perfect numbers to Mersenne primes.
Euler–Fermat theorem, that $a φ (m) \equiv 1 (mod m)$ whenever $a$ is coprime to $m$ , and $φ$ is Euler's totient function
Euler's partition theorem equating the number of partitions with odd parts and the number of partitions with distinct parts. Also see Glaisher's theorem.
Euler's adding-up theorem in economics

Euler's laws

Euler's first law, the linear momentum of a body is equal to the product of the mass of the body and the velocity of its center of mass.
Euler's second law, the sum of the external moments about a point is equal to the rate of change of angular momentum about that point.

Other things named after Euler

2002 Euler (a minor planet)
AMS Euler typeface
Euler (software)
Euler acceleration or force
Euler Book Prize
Euler Medal, a prize for research in combinatorics
Euler programming language
Euler Society, an American group dedicated to the life and work of Leonhard Euler
Euler–Fokker genus
Project Euler
Leonhard Euler Telescope
Rue Euler (a street in Paris, France)^[3]
Euler Park (a public park in Lima, Peru)

Topics by field of study

Selected topics from above, grouped by subject.

Analysis: derivatives, integrals, and logarithms

Euler approximation – (see Euler's method)
Euler derivative (as opposed to Lagrangian derivative)
The Euler integrals of the first and second kind, namely the beta function and gamma function.
The Euler method, a method for finding numerical solutions of differential equations
- Semi-implicit Euler method
Euler's number $e \approx 2.71828$ , the base of the natural logarithm, also known as Napier's constant.
The Euler substitutions for integrals involving a square root.
Euler's summation formula, a theorem about integrals.
Cauchy–Euler equation (or Euler equation), a second-order linear differential equation
Euler–Maclaurin formula – relation between integrals and sums
Euler–Mascheroni constant or Euler's constant $γ \approx 0.577216$

Geometry and spatial arrangement

Euler angles defining a rotation in space.
Euler brick
Euler's line – relation between triangle centers
Euler operator – set of functions to create polygon meshes
Euler's rotation theorem
Euler spiral – a curve whose curvature varies linearly with its arc length
Euler squares, usually called Graeco-Latin squares.
Euler's theorem in geometry, relating the circumcircle and incircle of a triangle.
Euler's quadrilateral theorem, an extension of the parallelogram law to convex quadrilaterals
Euler–Rodrigues formulas concern Euler–Rodrigues parameters and 3D rotation matrices

Graph theory

Euler characteristic (formerly called Euler number) in algebraic topology and topological graph theory, and the corresponding Euler's formula $\scriptstyle \chi(S^2)=F-E+V=2$
Eulerian circuit, Euler cycle or Eulerian path – a path through a graph that takes each edge once
- Eulerian graph has all its vertices spanned by an Eulerian path
Euler class
Euler diagram – incorrectly, but more popularly, known as Venn diagrams, its subclass
Euler tour technique

Music

Euler–Fokker genus

Number theory

Euler's criterion – quadratic residues modulo by primes
Euler product – infinite product expansion, indexed by prime numbers of a Dirichlet series
Euler pseudoprime
Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.

Physical systems

Euler's Disk – a toy consisting of a circular disk that spins, without slipping, on a surface
Euler rotation equations, in rigid body dynamics.
Euler conservation equations in fluid dynamics.
Euler number (physics), the cavitation number in fluid dynamics.
Euler's three-body problem
Euler–Bernoulli beam equation, concerning the elasticity of structural beams.
Euler formula in calculating the buckling load of columns.
Euler–Tricomi equation – concerns transonic flow
Euler integrals (thermodynamics) - Gives relationship between extensive variables in Thermodynamics^[4]

Polynomials

Euler's homogeneous function theorem, a theorem about homogeneous polynomials.
Euler polynomials
Euler spline – splines composed of arcs using Euler polynomials^[5]

Notes

↑ Richeson, David S. (2008). Euler's Gem: The polyhedron formula and the birth of topology (illustrated ed.). Princeton University Press. p. 86. ISBN 978-0-691-12677-7.
↑ Edwards, C. H.; Penney, David E. (2004). Differential equations and boundary value problems. 清华大学出版社. p. 443. ISBN 978-7-302-09978-9.
↑ de Rochegude, Félix (1910). Promenades dans toutes les rues de Paris [Walks along all of the streets in Paris] (VIII^e arrondissement ed.). Hachette. p. 98.
↑ "The Euler equation in thermodynamics" (blog). March 2013.
↑ Schoenberg (1973). "bibliography" (PDF). University of Wisconsin.

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[1] Richeson, David S. (2008). Euler's Gem: The polyhedron formula and the birth of topology (illustrated ed.). Princeton University Press. p. 86. ISBN 978-0-691-12677-7.

[2] Edwards, C. H.; Penney, David E. (2004). Differential equations and boundary value problems. 清华大学出版社. p. 443. ISBN 978-7-302-09978-9.

[3] Rochegude, Félix (1910). Promenades dans toutes les rues de Paris [Walks along all of the streets in Paris] (VIII^e arrondissement ed.). Hachette. p. 98.

[4] "The Euler equation in thermodynamics" (blog). March 2013.

[5] Schoenberg (1973). "bibliography" (PDF). University of Wisconsin.