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Difference between logarithm and harmonic series
{\displaystyle \log _{e}(x)} . Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase
Euler's_constant
Base of natural logarithms
after Euler. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli introduced the constant while
E_(mathematical_constant)
Complex exponential in terms of sine and cosine
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric
Euler's_formula
Mathematical equation linking e, i and π
π (π = 3.14159...), the fundamental circle constant The number e (e = 2.71828...), also known as Euler's number, which occurs widely in mathematical
Euler's_identity
Special constant related to the exponential integral
In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by δ {\displaystyle \delta } , appears in integral evaluations and as a value
Gompertz_constant
Swiss mathematician (1707–1783)
Louhivaara, I. S.; Winkler, J., eds. (May 1983). Zum Werk Leonhard Eulers: Vorträge des Euler-Kolloquiums im Mai 1983 in Berlin (PDF). Birkhäuser Verlag. doi:10
Leonhard_Euler
Fixed number that has received a name
more digits of π is a world record pursuit. Euler's number e, also known as the exponential growth constant, appears in many areas of mathematics, and
Mathematical_constant
Analytic function in mathematics
k)^{n}}{k}}\right)-{\frac {(\ln m)^{n+1}}{n+1}}\right)}.} The constant term γ0 is the Euler–Mascheroni constant. For all s ∈ C {\displaystyle \mathbb {C} } , s ≠
Riemann_zeta_function
Nearest integers from a number
into two sequences via the floor function. There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g. γ =
Floor_and_ceiling_functions
Number of integers coprime to and less than n
{(\log n)^{\frac {2}{3}}}{n}}\right)} (where γ is the Euler–Mascheroni constant). In 1965 P. Kesava Menon proved ∑ gcd ( k , n ) = 1 1 ≤ k ≤ n
Euler's_totient_function
Constants in the zeta function's Laurent series expansion
The constant γ 0 = γ = 0.577 … {\displaystyle \gamma _{0}=\gamma =0.577\dots } is known as the Euler–Mascheroni constant. The Stieltjes constants are
Stieltjes_constants
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or
List of mathematical constants
List_of_mathematical_constants
and sums Euler–Mascheroni constant or Euler's constant γ ≈ 0.577216 Integration using Euler's formula Euler summation Euler–Boole summation Euler angles
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Mathematical constant
mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to special functions like
Glaisher–Kinkelin_constant
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 constant
analysis and number theory, Somos' quadratic recurrence constant or simply Somos' constant is a constant defined as an expression of infinitely many nested
Somos' quadratic recurrence constant
Somos'_quadratic_recurrence_constant
arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10
List_of_numbers
Mathematical constant
{1}{p}}\right],} where γ {\displaystyle \gamma } is the Euler–Mascheroni constant. Mertens' second theorem establishes that the limit exists. The
Meissel–Mertens_constant
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
In mathematics, the lemniscate constant ϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its
Lemniscate_constant
Number, approximately 3.14
The number π (/paɪ/ ; spelled out as pi) is a mathematical constant that is the ratio of a circle's circumference to its diameter. It is approximately
Pi
In mathematics, a non-algebraic number
algebraically independent. The Euler–Mascheroni constant γ: In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes γ/4) contains
Transcendental_number
Unicode block
is the constant most usually known as "Euler's constant", Euler's number is also a constant. For more on the notation of those two constants, see their
Letterlike_Symbols
Leonhard Euler Faraday constant – Michael Faraday Feigenbaum constants – Mitchell Feigenbaum Fermi coupling constant – Enrico Fermi Gauss's constant – Carl
List of scientific constants named after people
List_of_scientific_constants_named_after_people
French mathematician
important results in the domain of factorial series. His representation of Euler's constant as a series of rational terms is well known. It was used in 1926 by
Joseph_Ser
Mathematical theorem
this relation it follows that the ring of differential operators with constant coefficients, generated by the Di, is commutative; but this is only true
Symmetry of second derivatives
Symmetry_of_second_derivatives
Rational numbers in a reciprocal logarithm
|}G_{n}{\big |}}{n}}=\gamma ,\end{aligned}}} where γ = 0.5772156649... is Euler's constant. These results are very old, and their history may be traced back to
Gregory_coefficients
Constant equal to twice pi
Leonhard Euler initially used the single letter π to denote the constant 6.28... in his 1727 Essay Explaining the Properties of Air. Euler would later
Tau_(mathematics)
Used to count, measure, and label
mathematical constant would later be named Euler's number (e). Irrational numbers began to be studied systematically in the 18th century, with Leonhard Euler who
Number
Negative integer two units from the origin in mathematics
August 21, 2025. Havil, J. (2003). Gamma: Exploring Euler's Constant [Gamma: Exploring Euler's Constant]. Princeton, New Jersey: Princeton University Press
−2
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
compressible Euler equations". The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Are Euler's constant γ {\displaystyle \gamma } and Catalan's constant G {\displaystyle G} irrational? Are they transcendental? Is Apéry's constant ζ (
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Sum of the inverses of the positive cubes
Apéry's constant is a mathematical constant, defined as the infinite sum of the reciprocals of the cubes of the positive integers. In symbols, ζ ( 3 )
Apéry's_constant
Italian mathematician (1750–1800)
only a compass (Mohr–Mascheroni theorem). He also calculated the Euler–Mascheroni constant to 32 decimal places. Lorenzo Mascheroni was born on 13 May 1750
Lorenzo_Mascheroni
Letter of several colonial Mayan alphabets in the Latin script, based on the digit 3
Letter Ɛ. – Black: Tresillo. – Green: Turned digit 3. – Red: Outdated Euler constant symbol. (Green and red characters shown only for fonts in which they
Tresillo_(letter)
generalized-Euler-constant function and its derivative". arXiv:0808.0410 [math.NT]. Sondow, J (2005). "Double integrals for Euler's constant and ln 4/π
Hadjicostas's_formula
Second-order partial differential equation describing motion of mechanical system
these into the Euler–Lagrange equation, we obtain d d x y ′ ( x ) 1 + ( y ′ ( x ) ) 2 = 0 y ′ ( x ) 1 + ( y ′ ( x ) ) 2 = C = constant ⇒ y ′ ( x ) = C
Euler–Lagrange_equation
Description of the orientation of a rigid body
simple form using Euler angles in the moving frame. Also the Euler's rigid body equations are simpler because the inertia tensor is constant in that frame
Euler_angles
Topics referred to by the same term
Martini–Henry cartridge The first three digits in the decimal expansion of Euler's constant, .5772156649... This disambiguation page lists articles associated
.577
Numbers expressible as integrals of algebraic functions
conjectured that 1 / π {\displaystyle 1/\pi } , Euler's number e {\displaystyle e} and the Euler–Mascheroni constant γ {\displaystyle \gamma } are not periods
Period_(number_theory)
Problem of deciding whether an expression equals zero
(January 1988). "Numerical Results on the Transcendence of Constants Involving π, e, and Euler's Constant" (PDF). Mathematics of Computation. 50 (20): 275–281
Constant_problem
Sum of inverse squares of natural numbers
{\displaystyle n\rightarrow \infty } . Havil, J. (2003), Gamma: Exploring Euler's Constant, Princeton, New Jersey: Princeton University Press, pp. 37–42 (Chapter
Basel_problem
Third letter of the Greek alphabet
Hermite constant The Euler’s Constant also known as Euler–Mascheroni constant ≈ 0.57721566490153286 Stieltjes constants Chvátal–Sankoff constants The lowercase
Gamma
Special mathematical function
\gamma } being Euler's constant, ϖ {\displaystyle \varpi } being the Lemniscate constant and G {\displaystyle G} being Catalan's constant. The last identity
Dirichlet_beta_function
Theorem on the existence of finite sets of integers >1 whose reciprocals sum to 1
known that, for this to be true, b {\displaystyle b} must be at least Euler's constant e {\displaystyle e} . Ernie Croot proved the conjecture as part of
Erdős–Graham_problem
Mathematical constant
mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, is a mathematical constant, which arises in the theory of random
Golomb–Dickman_constant
Ability of a body to store an electrical charge
plates, in square meters; ε 0 {\textstyle \varepsilon _{0}} is the electric constant ( ε 0 ≈ 8.854 × 10 − 12 F ⋅ m − 1 {\textstyle \varepsilon _{0}\approx
Capacitance
Characteristic time in a system
In physics and engineering, the time constant, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input
Time_constant
Mathematical constant in number theory
g. ln(2) and ln(3) The Euler-Mascheroni constant γ Apéry's constant ζ(3) The Feigenbaum constants δ and α Khinchin's constant itself (which would mean
Khinchin's_constant
Number that is not a ratio of integers
irrational and even transcendental. The question about the irrationality of Euler's constant γ is a long standing open problem in number theory. Other important
Irrational_number
Summation formula
infinity but the difference between them goes to a limit, the Euler–Mascheroni constant, γ ≈ 0.5772... This gives: ∫ 1 n ( 1 ⌊ x ⌋ − 1 x ) d x = ∫ 1 ∞
Euler–Maclaurin_formula
Infinite products of functions indexed by primes
order of increasing primes. Other Euler products for known constants include: The Hardy–Littlewood twin prime constant: ∏ p > 2 ( 1 − 1 ( p − 1 ) 2 ) =
Euler_product
Graphemes for various number systems
represent mathematical constants: U+210E ℎ PLANCK CONSTANT, the U+210F ℏ PLANCK CONSTANT OVER TWO PI, and U+2107 ℇ EULER CONSTANT (of unknown significance)
Numerals_in_Unicode
Graph in which every two vertices are adjacent
{\displaystyle w_{n+2}=n!e_{n}=\lfloor en!\rfloor ,} where e refers to Euler's constant, and e n = ∑ k = 0 n 1 k ! . {\displaystyle e_{n}=\sum _{k=0}^{n}{\frac
Complete_graph
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
than some given x is (eγ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm. It is widely believed
List of Mersenne primes and perfect numbers
List_of_Mersenne_primes_and_perfect_numbers
Indian mathematician (1887–1920)
developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant up to 15 decimal places. His peers at the time said they "rarely
Srinivasa_Ramanujan
Family of solutions to related differential equations
{\text{ is a negative integer,}}\end{cases}}} where γ is the Euler–Mascheroni constant (0.5772...). For the second case (where α {\displaystyle \alpha
Bessel_function
German meteorologist and physicist
synoptic meteorology. In 1824 he developed a new method to compute the Euler constant numerically. He died on 17 May 1834 in Leipzig and was buried in the
Heinrich_Wilhelm_Brandes
Conjecture on zeros of the zeta function
{n}{\log \log n}}} for infinitely many n, where φ(n) is Euler's totient function and γ is Euler's constant. Ribenboim remarks that: "The method of proof is interesting
Riemann_hypothesis
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
Mathematical model in quantum mechanics
\left({\frac {4\pi \hbar \,e^{2(1-\gamma )}}{L\,p_{0}}}\right)} where γ is Euler's constant. The quantum mechanical entropic uncertainty principle states that
Particle_in_a_box
Computer program
mathematical constant with theoretical accuracy limited only by computing time and available storage space. It was originally developed to calculate the Euler-Mascheroni
Y-cruncher
Infinitely many prime numbers exist
paper of his former proof. Havil, Julian (2003). Gamma: Exploring Euler's Constant. Princeton University Press. pp. 28–29. ISBN 0-691-09983-9. Furstenberg
Euclid's_theorem
Mathematical constant
_{n=1}^{\infty }{\frac {1}{2^{2n-1}(2n+1)}}\zeta (2n)=1-\ln 2.} (γ is the Euler–Mascheroni constant and ζ Riemann's zeta function.) ln 2 = 2 3 + 1 2 ∑ k = 1 ∞ (
Natural_logarithm_of_2
Problem in physics and astronomy
yield a net force and torque on the particle. Nevertheless, Euler's problem has a second constant of motion C = r 1 2 r 2 2 d θ 1 d t d θ 2 d t + 2 a ( μ
Euler's_three-body_problem
alternating sum of vertices, edges and faces equals a constant: V − E + F = 2. This constant, χ, is the Euler characteristic of the sphere. The study and generalization
Contributions of Leonhard Euler to mathematics
Contributions_of_Leonhard_Euler_to_mathematics
German mathematician (1808–1878)
first mathematicians to use the symbol γ {\displaystyle \gamma } for Euler's constant when he published his 1837 paper. He is best known for his discovery
Carl_Anton_Bretschneider
Fast summation method in mathematics
the classical constants e, π , {\displaystyle \pi ,} the Euler constant γ , {\displaystyle \gamma ,} the Catalan and the Apéry constants, such higher transcendental
FEE_method
Probability distribution
{\frac {\exp(-1/x)}{x(1+x)}},} where γ {\displaystyle \gamma } is Euler's constant. A similar approximation of p ( x ; μ , c ) {\displaystyle p(x;\mu
Landau_distribution
Wobble of the axis of rotation
is a movement of a rotational axis such that the first Euler angle (or precession) is constant. Astronomers usually make a distinction between precession
Nutation
Integers occurring in the coefficients of the Taylor series of 1/cosh t
Dirichlet beta function Euler–Mascheroni constant Jha, Sumit Kumar (2019). "A new explicit formula for Bernoulli numbers involving the Euler number". Moscow Journal
Euler_number
on the level of numeric and boolean constants. Thus, besides the traditional numeric and logical constants, Euler introduces several added types: Reference
Euler_(programming_language)
Mathematical function, inverse of an exponential function
converges (i.e. gets arbitrarily close) to a number known as the Euler–Mascheroni constant γ = 0.5772.... This relation aids in analyzing the performance
Logarithm
Concept in algebraic number theory
functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais. Ramanujan's constant is the transcendental
Heegner_number
Function that quantifies how near a number is to being rational
irrationality measure for Liouville numbers and conditional measures for Euler's constant". arXiv:math/0307308. Chudnovsky, G. V. (1982). "Hermite-padé approximations
Irrationality_measure
Concept in genetics
}{m}}{\text{ if }}mN_{e}\gg 1\end{cases}}} where γ {\displaystyle \gamma } is Euler's constant. The first approximation represents the waiting time until the first
Genetic_drift
Measure of change in amplitude and phase of a wave
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates
Propagation_constant
Continuous probability distribution, named after Benjamin Gompertz
γ is the Euler constant: γ = − ψ ( 1 ) = 0.577215... {\displaystyle {\begin{aligned}{\text{ where }}&\gamma {\text{ is the Euler constant: }}\,\!\\&\gamma
Gompertz_distribution
Characters encoded solely to maintain round-trip convertibility with other standards
LUNATE SIGMA SYMBOL Mathematical constants (3): U+2107 ℇ EULER CONSTANT, U+210E ℎ PLANCK CONSTANT, U+210F ℏ PLANCK CONSTANT OVER TWO PI Currency symbols (2):
Unicode compatibility characters
Unicode_compatibility_characters
Multiplicative function in number theory
{\mu (n)\ln ^{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
Möbius_function
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)
Divergent sum of positive unit fractions
logarithm and γ ≈ 0.577 {\displaystyle \gamma \approx 0.577} is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic
Harmonic_series_(mathematics)
Iterative algorithm on numbers
= 8532 8532 – 2358 = 6174 7641 – 1467 = 6174 6174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10)
Kaprekar's_routine
Mathematical function
(z)} . Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields
Digamma_function
Algorithmic runtime requirements for common math procedures
This table gives the complexity of computing approximations to the given constants to n {\displaystyle n} correct digits. Algorithms for number theoretical
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Speed of electromagnetic waves
for "constant" or the Latin celeritas (meaning 'swiftness, celerity'). The "c" was used for "celerity" meaning a velocity in books by Leonhard Euler and
Speed_of_light
Mathematical formula in harmonic analysis
_{n=1}^{X}d(n)-X\log X-(2\gamma -1)X=O(X^{1/2})} where γ {\displaystyle \gamma } is Euler's constant ≈ 0.57721566. Gauss’ circle problem concerns the average size of r
Voronoi_formula
Varying methods used to calculate pi
Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning
Approximations_of_pi
Force needed to pull a spring grows linearly with distance
linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small
Hooke's_law
Differential equation that is linear with respect to the unknown function
of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions
Linear_differential_equation
Increasing sequence of reduced fractions
(x)} is the Riemann zeta function and γ {\displaystyle \gamma } is Euler's constant. The sum of all Farey fractions of order n is half the number of elements:
Farey_sequence
Shape with same width in all directions
bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are
Curve_of_constant_width
Function in analytic number theory
27 (2000), 29–34. Sondow, Jonathan (2005). "Double integrals for Euler's constant and ln 4/π and an analog of Hadjicostas's formula". Amer. Math. Monthly
Dirichlet_eta_function
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
Antenna consisting of two rod-shaped conductors
57721566 … {\displaystyle \ \gamma _{e}=0.57721566\ \ldots \ } is Euler's constant. There is an equivalent alternate form favored by some authors that
Dipole_antenna
Approximation of the definite integral of a function
algorithms (see also GNU Scientific Library) From Lobatto Quadrature to the Euler constant e Gaussian Quadrature Rule of Integration – Notes, PPT, Matlab, Mathematica
Gaussian_quadrature
German mathematician
Maier's matrix method Maier's theorem Lagarias, Jeffrey (2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical
Helmut_Maier
Mathematical function
}\left({\frac {z}{n}}-\log \left(1+{\frac {z}{n}}\right)\right),} where γ is Euler's constant. Substituting ( 2 i t + 1 ) / 4 {\displaystyle (2it+1)/4} for z and
Riemann–Siegel_theta_function
Equations that describe the behavior of a physical system
variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to
Equations_of_motion
Mathematical function
(a)+(1-\gamma )\rho (a-1)(x/\log x)+O(x/{(\log x)}^{2})} where γ is Euler's constant. The main purpose of the Dickman–de Bruijn function is to estimate
Dickman_function
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EULERS CONSTANT
EULERS CONSTANT
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