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Number of integers coprime to and less than n
log e ( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle
Euler's_totient_function
Mathematical function
In mathematics, the Euler function is given by ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad
Euler_function
Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. These include functions, equations, formulas, identities
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Extension of the factorial function
absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.) The relationship to the factorial
Gamma_function
Swiss mathematician (1707–1783)
notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal
Leonhard_Euler
Complex exponential in terms of sine and cosine
Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the
Euler's_formula
Integers occurring in the coefficients of the Taylor series of 1/cosh t
{\displaystyle \cosh(t)} is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely E n = 2 n E n ( 1 2
Euler_number
Number of partitions of an integer
exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal
Partition function (number theory)
Partition_function_(number_theory)
Difference between logarithm and harmonic series
\mathrm {d} x.\end{aligned}}} Here, ⌊·⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: 0.57721 56649 01532
Euler's_constant
Second-order partial differential equation describing motion of mechanical system
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose
Euler–Lagrange_equation
Approach to finding numerical solutions of ordinary differential equations
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary
Euler_method
Mathematical function
{\displaystyle x=2\pi i\tau } in Euler Pentagonal number theorem with the definition of eta function. Another way to see the Eta function is through the following
Dedekind_eta_function
Mathematical equation linking e, i and π
Euler's identity (also known as Euler's equation) is the equality e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} where e {\displaystyle e} is Euler's number
Euler's_identity
Polynomial sequence
series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular
Bernoulli_polynomials
Infinite products of functions indexed by primes
proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. In general, if
Euler_product
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Function studied by Ramanujan
(z),} where ϕ {\displaystyle \phi } is the Euler function, η {\displaystyle \eta } is the Dedekind eta function, Δ ( z ) {\displaystyle \Delta (z)} is the
Ramanujan_tau_function
Theorem in number theory
In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that ∏ n = 1 ∞ ( 1 −
Pentagonal_number_theorem
Meromorphic function on the complex plane
an L-function via analytic continuation, is called an L-series. Fundamental subclasses of L-functions were built on the work of Leonhard Euler (which
L-function
Euler Mathematical Toolbox (or EuMathT; formerly Euler) is a free and open-source numerical software package. It contains a matrix language, a graphical
Euler_Mathematical_Toolbox
Use of a Dirichlet series expansion to calculate the complex function
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations
Proof of the Euler product formula for the Riemann zeta function
Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Base of natural logarithms
sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with other numbers named after Euler. Alternatively
E_(mathematical_constant)
Function with a multiplicative scaling behaviour
the exponential function x ↦ e x {\displaystyle x\mapsto e^{x}} are not homogeneous. Roughly speaking, Euler's homogeneous function theorem asserts that
Homogeneous_function
Curve whose curvature changes linearly
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the
Euler_spiral
Graphical set representation involving overlapping shapes
An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining
Euler_diagram
and terminology. Euler introduced much of the mathematical notation in use today, such as the notation f(x) to describe a function and the modern notation
Contributions of Leonhard Euler to mathematics
Contributions_of_Leonhard_Euler_to_mathematics
Topological invariant in mathematics
algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant
Euler_characteristic
Description of the orientation of a rigid body
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They
Euler_angles
Odd composite number which passes the given congruence
In mathematics, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and a ( n − 1 ) / 2 ≡ ± 1 ( mod n ) {\displaystyle
Euler_pseudoprime
Theorem on modular exponentiation
denotes Euler's totient function; that is a φ ( n ) ≡ 1 ( mod n ) . {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.} In 1736, Leonhard Euler published
Euler's_theorem
Summation formula
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate
Euler–Maclaurin_formula
Mathematical function
}} This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of
Ramanujan_theta_function
Numerical method for ordinary differential equations
numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the
Backward_Euler_method
Special functions of several complex variables
Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results
Theta_function
Method for load calculation in construction
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which
Euler–Bernoulli_beam_theory
Number, approximately 3.14
can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula: e i φ = cos φ + i sin φ , {\displaystyle
Pi
Topics referred to by the same term
beta function, also called the Euler beta function or the Euler integral of the first kind, is a special function in mathematics. Beta function may also
Beta function (disambiguation)
Beta_function_(disambiguation)
Generalizations of the Riemann zeta function
Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler: ∑ n = 1 ∞ H n
Multiple_zeta_function
Mathematical function, denoted exp(x) or e^x
exponential function can also be computed with continued fractions. A continued fraction for ex can be obtained via an identity of Euler: e x = 1 + x
Exponential_function
A prime p divides a^p–a for any integer a
{\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}},} where φ(n) denotes Euler's totient function (which counts the integers from 1 to n that are coprime to n).
Fermat's_little_theorem
Functions of an angle
to that of the proof of Euler's formula (see § Euler's formula and the exponential function above). One can also use Euler's formula for expressing all
Trigonometric_functions
Conjecture on zeros of the zeta function
convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is
Riemann_hypothesis
Mathematical function
digamma function: Γ ′ ( z ) Γ ( z ) = ψ ( z ) {\displaystyle {\frac {\Gamma '(z)}{\Gamma (z)}}=\psi (z)} . Euler's product formula for the gamma function, combined
Digamma_function
Use of complex numbers to evaluate integrals
integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric
Integration using Euler's formula
Integration_using_Euler's_formula
Problem in number theory on equal totients
mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ ( n ) {\displaystyle \varphi (n)}
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Logarithm to the base of the mathematical constant e
{\displaystyle |x|\leq 1} and x ≠ − 1. {\displaystyle x\neq -1.} Leonhard Euler, disregarding x ≠ − 1 {\displaystyle x\neq -1} , nevertheless applied this
Natural_logarithm
Integral of the Gaussian function, equal to sqrt(π)
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German
Gaussian_integral
In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function. By induction, it is easy to show that independent
Euler_measure
Infinite series summing alternating 1 and -1 terms
+ 1 + 1 − 1 − 1 + ⋯ occurs in Euler's 1775 treatment of the pentagonal number theorem as the value of the Euler function at q = 1. The power series most
Grandi's_series
Index of articles associated with the same name
In mathematics, there are two types of Euler integral: The Euler integral of the first kind is the beta function B ( z 1 , z 2 ) = ∫ 0 1 t z 1 − 1 ( 1
Euler_integral
Odd composite number which passes the given congruence
literature (for example,), an Euler–Jacobi pseudoprime as defined above is often called simply an Euler pseudoprime. function EulerJacobiTest(k) a = 2 if k
Euler–Jacobi_pseudoprime
Method in Itô calculus
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential
Euler–Maruyama_method
Generalized function whose value is zero everywhere except at zero
which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics, is the rescaled Airy function ε − 1 / 3 Ai ( x ε − 1 / 3 ) . {\displaystyle
Dirac_delta_function
Mathematical concept
result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be
Infinite_product
Summation method for some divergent series
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series
Euler_summation
Differential calculus on function spaces
integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of
Calculus_of_variations
Functions such that f(–x) equals f(x) or –f(x)
function and an odd function. The concept of even and odd functions appears to date back to the early 18th century, with Leonhard Euler playing a significant
Even_and_odd_functions
Analytic function that does not satisfy a polynomial equation
The hyperbolic logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised
Transcendental_function
Arithmetic function related to the divisors of an integer
lists a few identities involving the divisor functions Euler's totient function, Euler's phi function Refactorable number Table of divisors Unitary divisor
Divisor_function
Decomposition of an integer as a sum of positive integers
multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal
Integer_partition
Mathematical concept in number theory
Euler product in its region of convergence. Analogous partial products can be formed for Dirichlet L-functions and other Dirichlet series with Euler products
Partial_Euler_Product
Special function defined by an integral
\end{aligned}}} The Euler spiral, also known as a Cornu spiral or clothoid, is the curve generated by a parametric plot of S(t) against C(t). The Euler spiral was
Fresnel_integral
Study of space and shapes locally given by a convergent power series
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem
Geometric_function_theory
Special constant related to the exponential integral
constant or Euler–Gompertz constant, denoted by δ {\displaystyle \delta } , appears in integral evaluations and as a value of special functions. It is named
Gompertz_constant
Fundamental trigonometric functions
elliptic functions Euler's formula Generalized trigonometry Hyperbolic function Lemniscate elliptic functions Law of sines List of periodic functions List
Sine_and_cosine
Transformation of a mathematical sequence
to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. The binomial
Binomial_transform
Number divisible only by 1 and itself
the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are
Prime_number
Association of one output to each input
that the function is f: S → S. The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler. However
Function_(mathematics)
Method of integration for rational functions
Euler substitution is a method for evaluating integrals of the form ∫ R ( x , a x 2 + b x + c ) d x , {\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\
Euler_substitution
Divergent series
the lines of Euler's reasoning, uses the relationship between the Riemann zeta function and the Dirichlet eta function η(s). The eta function is defined
1_+_2_+_3_+_4_+_⋯
Function defined by a hypergeometric series
{1}{2}};1;k^{2}\right).\end{aligned}}} The hypergeometric function is a solution of Euler's hypergeometric differential equation z ( 1 − z ) d 2 w d z
Hypergeometric_function
Mathematical approximation of a function
x are Euler numbers. The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions: sinh
Taylor_series
Course designed to prepare students for calculus
general logarithm, to an arbitrary positive base, Euler presents as the inverse of an exponential function. Then the natural logarithm is obtained by taking
Precalculus
Special mathematical function defined as sin(x)/x
{x^{2}}{n^{2}}}\right)} and is related to the gamma function Γ(x), as well as to Gauss' Pi function, through Euler's reflection formula: sin ( π x ) π x = 1 Γ
Sinc_function
Mathematical function with no sudden changes
In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies
Continuous_function
Summation method for some divergent series
{2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.} The periodic Euler functions modify these by a sign change depending on the parity of the integer
Euler–Boole_summation
Type of functional equation (mathematics)
the unknown function at a point to its values at nearby points. Many numerical methods for differential equations, for example the Euler method, involve
Differential_equation
Multiplicative function in number theory
^{2}n}{n}}=-2\gamma ,} where γ {\displaystyle \gamma } is Euler's constant. The Lambert series for the Möbius function is ∑ n = 1 ∞ μ ( n ) q n 1 − q n = q , {\displaystyle
Möbius_function
Family of solutions to related differential equations
function. Leonhard Euler in 1736, found a link between other functions (now known as Laguerre polynomials) and Bernoulli's solution. Euler also introduced
Bessel_function
Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous function Derivative
List_of_calculus_topics
Special mathematical function
(s)} where Γ ( s ) {\displaystyle \Gamma (s)} is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842. For every odd positive
Dirichlet_beta_function
Hyperbolic analogues of trigonometric functions
Charles Edward. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100. Becker, Georg F. Hyperbolic functions. Read Books, 1931
Hyperbolic_functions
Matrix of partial derivatives of a vector-valued function
calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Short exact sequence of sheaves on projective space
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of
Euler_sequence
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
model uses the Euler numerical method to model consumption. He created his consumption theory in response to the Lucas critique. Using Euler equations to
Random walk model of consumption
Random_walk_model_of_consumption
Multivalued function in mathematics
exponential function. The function is named after Johann Lambert, who considered a related problem in 1758. Building on Lambert's work, Leonhard Euler described
Lambert_W_function
Modification of the Euler method for solving Hamilton's equations
semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method
Semi-implicit_Euler_method
{Q(x)Q(x^{5})}{Q(x^{3})^{2}}}.} The functions G and H turn up in the Rogers–Ramanujan identities, and the function Q is the Euler function, which is closely related
Hard_hexagon_model
Book by Leonhard Euler
quadrature of the hyperbola and the trigonometric functions to the arc-length of the circle. Euler accomplished this feat by introducing exponentiation
Introductio in analysin infinitorum
Introductio_in_analysin_infinitorum
Mathematical theorem
a long history. The list of unsuccessful proposed proofs started with Euler's, published in 1740, although already in 1721 Bernoulli had implicitly assumed
Symmetry of second derivatives
Symmetry_of_second_derivatives
Conditions for switching order of integration in calculus
Cavalieri's principle, which was used by Leonhard Euler. More formally, the theorem states that if a function is Lebesgue integrable on a rectangle X × Y {\displaystyle
Fubini's_theorem
Branch of mathematics
integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of
Calculus
Instantaneous rate of change (mathematics)
quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input
Derivative
Number equal to the sum of its proper divisors
Two millennia later, Leonhard Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether
Perfect_number
Ordinary differential equation
in 1768. Let y(n)(x) be the nth derivative of the unknown function y(x). Then a Cauchy–Euler equation of order n has the form a n x n y ( n ) ( x ) + a
Cauchy–Euler_equation
Diagram that shows all possible logical relations between a collection of sets
as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler in 1768 (Letters to a German Princess). The idea was popularised by Venn
Venn_diagram
Statement relating differentiable symmetries to conserved quantities
change in I, at least up to first order. This principle results in the Euler–Lagrange equations, d d t ( ∂ L ∂ q ˙ ) = ∂ L ∂ q . {\displaystyle {\frac
Noether's_theorem
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