List of topics named after Leonhard Euler

In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. These include functions, equations, formulas, identities, numbers (single or sequence), or other mathematical entities. Many of these entities have been given simple yet 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 view of the high number of properties and theorems discovered by Euler, many of them are conventionally named after the first person to have proved them after him.

Conjectures

Euler's sum of powers conjecture – disproved for exponents 4 and 5 during the 20th century; unsolved for higher exponents
Euler's Graeco-Latin square conjecture – proved to be true for ⁠ $n=6$ ⁠ and disproved otherwise, during the 20th century

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 may 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

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 ODE with variable coefficients. Its second-order version can emerge from Laplace's equation in polar coordinates.
Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams.
Euler's differential equation, a first order nonlinear ordinary differential equation

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–Lagrange equation, a second-order PDE emerging from minimization problems in calculus of variations.
Euler–Arnold equation, describes the evolution of a velocity field when the Lagrangian flow is a geodesic in a group of smooth transformations.

Formulas

Euler's formula, $e ix = cos x + i sin x$
Euler's polyhedral formula for planar graphs or polyhedra: $v - e + f = 2$ , a special case of the Euler characteristic in topology
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 reflection formula, reflection formula for the gamma function
Local Euler characteristic formula

Functions

The Euler function, a modular form that is a prototypical q-series.
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–Riemann zeta function

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.

Numbers

Euler's number, $e=2.71828\dots$ , the base of the natural logarithm
Euler's idoneal numbers, a set of 65 or possibly 66 or 67 integers with special properties
Euler numbers, integers occurring in the coefficients of the Taylor series of 1/cosh t
Eulerian numbers count certain types of permutations.
Euler number (physics), the cavitation number in fluid dynamics.
Euler number (algebraic topology) – now, Euler characteristic, classically the number of vertices minus edges plus faces of a polyhedron.
Euler number (3-manifold topology) – see Seifert fiber space
Lucky numbers of Euler
Euler's constant gamma $\gamma =0.57721\dots$ , also known as the Euler–Mascheroni constant
Eulerian integers, more commonly called Eisenstein integers, the algebraic integers of form $a + bω$ where $ω$ is a complex cube root of 1.
Euler–Gompertz constant

Theorems

Euler's homogeneous function theorem – A homogeneous function is a linear combination of its partial derivatives
Euler's infinite tetration theorem – About the limit of iterated exponentiation
Euler's rotation theorem – Movement with a fixed point is rotation
Euler's theorem (differential geometry) – Orthogonality of the directions of the principal curvatures of a surface
Euler's theorem in geometry – On distance between centers of a triangle
Euler's quadrilateral theorem – Relation between the sides of a convex quadrilateral and its diagonals
Euclid–Euler theorem, characterizing even perfect numbers
Euler's theorem, on modular exponentiation
Euler's partition theorem relating the product and series representations of the Euler function Π(1 − xⁿ)
Goldbach–Euler theorem, stating that sum of 1/(k − 1), where k ranges over positive integers of the form mⁿ for m ≥ 2 and n ≥ 2, equals 1
Gram–Euler theorem

Laws

Euler's first law, the sum of the external forces acting on a rigid body is equal to the rate of change of linear momentum of the body.
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

2002 Euler, a minor planet
Euler (crater), a lunar impact crater
AMS Euler typeface
Euler Mathematical Toolbox computer software
Euler Book Prize, an annual prize for mathematics books
Euler Lecture, an annual lecture at the University of Potsdam
Euler Medal, a prize for research in combinatorics
Leonhard Euler Gold Medal, a prize for outstanding results in mathematics and physics
Euler (programming language), an Algol derivative
Euler Society, an American group dedicated to the life and work of Leonhard Euler
Euler Committee of the Swiss Academy of Sciences, a group dedicated to publishing the Euler's scientific productions
Euler–Fokker genus, a musical scale
Project Euler, a collection of programming puzzles
Leonhard Euler Telescope, a Swiss-operated telescope in Chile
Rue Euler [fr], a street in Paris, France
EulerOS, a CentOS Linux based operating system
French submarine Euler, an early French submarine
Euler square, a concept in combinatorics
Euler top, a special case of a rotating rigid body in classical mechanics

Topics by field of study

Selected topics from above, grouped by subject, and additional topics from the fields of music and physical systems

Analysis: derivatives, integrals, and logarithms

Euler approximation – (see Euler's method)
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
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
Cauchy–Euler operator
Euler–Maclaurin formula – relation between integrals and sums
Euler–Mascheroni constant or Euler's constant $γ \approx 0.577216$
Integration using Euler's formula
Euler summation
Euler–Boole summation

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 filter
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 formula concerning Euler–Rodrigues parameters and 3D rotation matrices
Cramer–Euler paradox
Euler calculus
Euler sequence
Gram–Euler theorem
Euler measure

Graph theory

Euler characteristic (formerly called Euler number) in algebraic topology and topological graph theory, and the corresponding Euler's formula ${\textstyle \chi (S^{2})=F-E+V=2}$
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 – popularly called "Venn diagrams", although some use this term only for a subclass of Euler diagrams.
Euler tour technique

Music

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–Jacobi pseudoprime
Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.
Euler system
Euler's factorization method

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–Lagrange equation
Euler–Tricomi equation – concerns transonic flow
Euler relations – Gives relationship between extensive variables in thermodynamics.
Eulerian observer – An observer "at rest" in spacetime, i.e. with 4-velocity perpendicular to spatial hypersurfaces.
Relativistic Euler equations
Euler top
Eulerian specification of the flow field
Euler nutation
Newton–Euler equations
d'Alembert–Euler condition
Euler acceleration or force

Polynomials

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

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, Charles Henry; Penney, David E.; Calvis, David (2008). Differential equations and boundary value problems. Pearson Prentice Hall. pp. 443 (微分方程及边值问题, 2004 edition). ISBN 978-0-13-156107-6.
↑ Weisstein, Eric W. "Totient Function". mathworld.wolfram.com. Retrieved 2025-01-31.
↑ Weisstein, Eric W. "Lucky Number of Euler". mathworld.wolfram.com. Retrieved 2025-01-31.
↑ 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.
↑ Evans, Charles R.; Smarr, Larry L.; Wilson, James R. (1986). "Numerical Relativistic Gravitational Collapse with Spatial Time Slices". Astrophysical Radiation Hydrodynamics. Vol. 188. pp. 491–529. doi:10.1007/978-94-009-4754-2_15. ISBN 978-94-010-8612-7. Retrieved March 27, 2021.
↑ Schoenberg (1973). "bibliography" (PDF). University of Wisconsin. Archived from the original (PDF) on 2011-05-22. Retrieved 2007-10-28.

[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, Charles Henry; Penney, David E.; Calvis, David (2008). Differential equations and boundary value problems. Pearson Prentice Hall. pp. 443 (微分方程及边值问题, 2004 edition). ISBN 978-0-13-156107-6.

[3] Weisstein, Eric W. "Totient Function". mathworld.wolfram.com. Retrieved 2025-01-31.

[4] Weisstein, Eric W. "Lucky Number of Euler". mathworld.wolfram.com. Retrieved 2025-01-31.

[5] 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.

[6] Evans, Charles R.; Smarr, Larry L.; Wilson, James R. (1986). "Numerical Relativistic Gravitational Collapse with Spatial Time Slices". Astrophysical Radiation Hydrodynamics. Vol. 188. pp. 491–529. doi:10.1007/978-94-009-4754-2_15. ISBN 978-94-010-8612-7. Retrieved March 27, 2021.

[7] Schoenberg (1973). "bibliography" (PDF). University of Wisconsin. Archived from the original (PDF) on 2011-05-22. Retrieved 2007-10-28.