In mathematics and physics, many topics are 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 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 an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.

Conjectures

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:

Ordinary differential equations

Partial differential equations

Formulas

• Euler's formula, • eix = 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: • 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

Identities

eiπ + 1 = 0 .

Numbers

e = 2.71828. .. , the base of the natural logarithm a + bω where ω is a complex cube root of 1.

Theorems

Laws

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, 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 • *Semi-implicit Euler method • *Euler–Maruyama method • *Backward Euler method • Euler's number • e ≈ 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 ≈ 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

Music

Number theory

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