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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
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
Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
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 value Γ ( 1 ) {\displaystyle
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
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Complex exponential in terms of sine and cosine
fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one
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_numbers
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
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)
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
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
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
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
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
2.71828...; base of natural logarithms
exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers
E_(mathematical_constant)
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
Functions of an angle
to that of the above proof of Euler's formula. One can also use Euler's formula for expressing all trigonometric functions in terms of complex exponentials
Trigonometric_functions
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
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
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
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_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
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)
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
Odd composite number which passes the given congruence
In number theory, an odd integer n is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base a, if a and n are coprime
Euler–Jacobi_pseudoprime
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
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 f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}
Gaussian_integral
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
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
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
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
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
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
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
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
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
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)
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
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
Complex-differentiable (mathematical) function
(}\exp(+iz)-\exp(-iz){\bigr )}} (cf. Euler's formula). The principal branch of the complex logarithm function log z {\displaystyle \log z} is holomorphic
Holomorphic_function
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
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
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
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
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
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
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
Branch of mathematics studying functions of a complex variable
prior. Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century
Complex_analysis
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
Mathematical function with no sudden changes
a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies
Continuous_function
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
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
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
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
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
Matrix of second derivatives
partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix
Hessian_matrix
Infinite series with alternating signs
Dirichlet eta function. Part of Euler's motivation for studying series related to 1 − 2 + 3 − 4 + ... was the functional equation of the eta function, which
1_−_2_+_3_−_4_+_⋯
Function equal to the product of its values on coprime factors
as a function of n {\displaystyle n} , where k {\displaystyle k} is a fixed integer φ ( n ) {\displaystyle \varphi (n)} : Euler's totient function, which
Multiplicative_function
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
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
Relationship between derivatives and integrals
differentiating a function (calculating its slopes, or rate of change at every point on its domain) with the concept of integrating a function (calculating
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
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
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_+_⋯
On the distribution of prime numbers in arithmetic progressions
\infty )} where φ {\displaystyle \varphi } is Euler's totient function. If we then define the error function E ( x ; q ) = max gcd ( a , q ) = 1 | π ( x
Elliott–Halberstam_conjecture
Problem in physics and astronomy
In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other
Euler's_three-body_problem
Operation in mathematical calculus
also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals
Integral
Notation of differential calculus
named after Joseph Louis Lagrange, although it was in fact invented by Euler and popularized by the former. In Lagrange's notation, a prime mark denotes
Notation_for_differentiation
Ordinary differential equation
In mathematics, an Euler–Cauchy equation, also known as a Cauchy–Euler equation, equidimensional equation, or Euler's equation, is a linear ordinary differential
Cauchy–Euler_equation
Quasilinear first-order ordinary differential equation
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a
Euler's equations (rigid body dynamics)
Euler's_equations_(rigid_body_dynamics)
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
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
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
EULER FUNCTION
EULER FUNCTION
Boy/Male
Christian, German, Norse, Polish, Scandinavian, Swedish
Peaceful Ruler; Forever; Alone; Ruler; All-ruler
Boy/Male
German
Powerful Ruler; Army Ruler
Boy/Male
Muslim
Ruler
Boy/Male
British, English
Wheel Ruler; Circle Ruler
Boy/Male
French, German, Irish
Dominant Ruler; Powerful Ruler
Boy/Male
Indian
Ruler
Boy/Male
Indian
Ruler
Boy/Male
American, Chinese, Christian, Danish, French, German, Norse, Scandinavian, Swedish
Ruler; Ruler of the People; Peaceful Ruler; All-ruler; Forever; Alone; Ever Ruler
Boy/Male
Danish, German, Swedish
Island Ruler; Ever Ruler
Boy/Male
Australian, Dutch, French, German, Italian, Latin, Swiss
Powerful Ruler; Dominant Ruler
Boy/Male
American, Australian, Danish, German
Powerful Ruler; Dominant Ruler
Boy/Male
Indian
Ruler
Boy/Male
French, German
Wise Ruler; Old Ruler; Long Term Ruler
Boy/Male
Muslim
Ruler
Boy/Male
Christian, German, Teutonic
Hard Working Ruler; Industrious Ruler; Home Ruler
Boy/Male
German, Teutonic
Hardworking Ruler; Home Ruler
Boy/Male
American, Czech, Danish, French, German, Scandinavian, Swedish
Honourable Ruler; Peaceful Ruler; All Ruler; Ever Ruler
Boy/Male
German, Swedish
Ever Ruler; Island Ruler
Boy/Male
American, British, English
Royal Ruler; King's Ruler
Boy/Male
American, Anglo, British, Christian, English, German
Wealthy Ruler; Rich Ruler
EULER FUNCTION
EULER FUNCTION
Boy/Male
Hindu
Name of a sage, From the heart
Girl/Female
Gaelic
Boy/Male
Hindu
Girl/Female
American, Bengali, Christian, Danish, English, Finnish, French, German, Greek, Gujarati, Hindu, Indian, Japanese, Kannada, Latin, Marathi, Sindhi, Swedish, Tamil, Telugu
A Wise Counsellor; Goddess Durga; Solitary; Advise; Counsel; Lord of Mind; Intellectual
Female
English
Pet form of English Nichole, NIKKI means "victor of the people."
Boy/Male
Christian & English(British/American/Australian)
Residence Name
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Unique; Lord Ganpati; One of Its Kind
Boy/Male
Indian, Sanskrit
Radiant; Beaming; Praising
Boy/Male
Tamil
From Sanskrit samit: someone who has got everything
Girl/Female
British, Danish, English, Latin
Mercy; Merciful
EULER FUNCTION
EULER FUNCTION
EULER FUNCTION
EULER FUNCTION
EULER FUNCTION
n.
A ruler of one division of a heptarchy.
n.
One who rules; one who exercises sway or authority; a governor.
n.
A chief or ruler of a deme or district in Greece.
a.
The office of ruler; rule; authority; government.
n.
A ruler or ruling power.
n.
A ruler; a governor; a prince.
n.
A ruler or governor.
n.
A straight or curved strip of wood, metal, etc., with a smooth edge, used for guiding a pen or pencil in drawing lines. Cf. Rule, n., 7 (a).
a.
Pertaining to Euler, a German mathematician of the 18th century.
n.
A long, flexble piece of wood sometimes used as a ruler.
n.
A ruler, or sovereign, of a Mohammedan state; specifically, the ruler of the Turks; the Padishah, or Grand Seignior; -- officially so called.
n.
A petty king; a ruler of little power or consequence.
n.
The mother and ruler of a family or of her descendants; a ruler by maternal right.
n.
A joint regent or ruler.
a.
One who rules or reigns; a governor; a ruler.
n.
A chief ruler; a potentate. [Obs.] Wyclif.
n.
A sole or supreme ruler; a sovereign; the highest ruler; an emperor, king, queen, prince, or chief.
a.
A suffix meaning a ruler, as in monarch (a sole ruler).
n.
A Mohammedan title for a ruler; a judge.
n.
One who pules; one who whines or complains; a weak person.