Euler's constant

Symbol: ConstGamma —

\gamma

— The constant gamma (0.577...)

The real number giving the limiting difference between the harmonic series and the natural logarithm, also known as Euler's constant or the Euler-Mascheroni constant.

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("39e0cb"),
    SymbolDefinition(ConstGamma, ConstGamma, "The constant gamma (0.577...)"),
    Description("The real number giving the limiting difference between the harmonic series and the natural logarithm, also known as Euler's constant or the Euler-Mascheroni constant."))

e876e8

\gamma \in \left[0.57721566490153286060651209008240243104215933593992 \pm 3.60 \cdot 10^{-51}\right]

TeX:

\gamma \in \left[0.57721566490153286060651209008240243104215933593992 \pm 3.60 \cdot 10^{-51}\right]

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("e876e8"),
    Formula(Element(ConstGamma, RealBall(Decimal("0.57721566490153286060651209008240243104215933593992"), Decimal("3.60e-51")))))

28bf9a

\gamma \notin \left\{ \frac{p}{q} : p \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, q \le {10}^{242080} \right\}

References:

J. Havil (2003): Exploring Euler's Constant. Princeton University Press. Page 97.

TeX:

\gamma \notin \left\{ \frac{p}{q} : p \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, q \le {10}^{242080} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("28bf9a"),
    Formula(NotElement(ConstGamma, SetBuilder(Div(p, q), Tuple(p, q), And(Element(p, ZZ), Element(q, ZZGreaterEqual(1)), LessEqual(q, Pow(10, 242080)))))),
    References("J. Havil (2003): Exploring Euler's Constant. Princeton University Press. Page 97."))

4644c0

\gamma = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{k} - \log\!\left(n\right)

TeX:

\gamma = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{k} - \log\!\left(n\right)

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`SequenceLimit`	$\lim_{n \to a} f\!\left(n\right)$	Limiting value of sequence
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("4644c0"),
    Formula(Equal(ConstGamma, SequenceLimit(Sub(Sum(Div(1, k), Tuple(k, 1, n)), Log(n)), n, Infinity))))

a1f1ec

\gamma = \lim_{s \to 1} \zeta\!\left(s\right) - \frac{1}{s - 1}

TeX:

\gamma = \lim_{s \to 1} \zeta\!\left(s\right) - \frac{1}{s - 1}

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("a1f1ec"),
    Formula(Equal(ConstGamma, ComplexLimit(Sub(RiemannZeta(s), Div(1, Sub(s, 1))), s, 1))))

cf3977

\gamma = -\Gamma'(1)

TeX:

\gamma = -\Gamma'(1)

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function

Source code for this entry:

Entry(ID("cf3977"),
    Formula(Equal(ConstGamma, Neg(Derivative(GammaFunction(z), Tuple(z, 1, 1))))))

d17d0b

\gamma = -\psi\!\left(1\right)

TeX:

\gamma = -\psi\!\left(1\right)

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("d17d0b"),
    Formula(Equal(ConstGamma, Neg(DigammaFunction(1)))))

818008

\gamma = 1 - \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right) - 1}{k}

TeX:

\gamma = 1 - \sum_{k=2}^{\infty} \frac{\zeta\!\left(k\right) - 1}{k}

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("818008"),
    Formula(Equal(ConstGamma, Sub(1, Sum(Div(Sub(RiemannZeta(k), 1), k), Tuple(k, 2, Infinity))))))

39fe5f

\gamma = -\int_{0}^{\infty} {e}^{-x} \log\!\left(x\right) \, dx

TeX:

\gamma = -\int_{0}^{\infty} {e}^{-x} \log\!\left(x\right) \, dx

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Exp`	${e}^{z}$	Exponential function
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("39fe5f"),
    Formula(Equal(ConstGamma, Neg(Integral(Mul(Exp(Neg(x)), Log(x)), Tuple(x, 0, Infinity))))))

a1ca3e

\gamma = -\int_{0}^{1} \log\!\left(\log\!\left(\frac{1}{x}\right)\right) \, dx

TeX:

\gamma = -\int_{0}^{1} \log\!\left(\log\!\left(\frac{1}{x}\right)\right) \, dx

Definitions:

Fungrim symbol	Notation	Short description
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Log`	$\log\!\left(z\right)$	Natural logarithm

Source code for this entry:

Entry(ID("a1ca3e"),
    Formula(Equal(ConstGamma, Neg(Integral(Log(Log(Div(1, x))), Tuple(x, 0, 1))))))

014c4e

\left|\gamma - \left(\frac{S}{I} - \frac{T}{{I}^{2}} - \log\!\left(n\right)\right)\right| \lt 24 {e}^{-8 n}\; \text{ where } \left(S, I, T\right) = \left(\sum_{k=0}^{5 n} \frac{H_{k} {n}^{2 k}}{{\left(k !\right)}^{2}}, \sum_{k=0}^{5 n} \frac{{n}^{2 k}}{{\left(k !\right)}^{2}}, \frac{1}{4 n} \sum_{k=0}^{2 n - 1} \frac{{\left(\left(2 k\right)!\right)}^{3}}{{\left(k !\right)}^{4} {8}^{2 k} {\left(2 n\right)}^{2 k}}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

References:

R. Brent and F. Johansson. A bound for the error term in the Brent-McMillan algorithm. Mathematics of Computation 2015, 84(295). DOI: 10.1090/S0025-5718-2015-02931-7

TeX:

\left|\gamma - \left(\frac{S}{I} - \frac{T}{{I}^{2}} - \log\!\left(n\right)\right)\right| \lt 24 {e}^{-8 n}\; \text{ where } \left(S, I, T\right) = \left(\sum_{k=0}^{5 n} \frac{H_{k} {n}^{2 k}}{{\left(k !\right)}^{2}}, \sum_{k=0}^{5 n} \frac{{n}^{2 k}}{{\left(k !\right)}^{2}}, \frac{1}{4 n} \sum_{k=0}^{2 n - 1} \frac{{\left(\left(2 k\right)!\right)}^{3}}{{\left(k !\right)}^{4} {8}^{2 k} {\left(2 n\right)}^{2 k}}\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`ConstGamma`	$\gamma$	The constant gamma (0.577...)
`Pow`	${a}^{b}$	Power
`Log`	$\log\!\left(z\right)$	Natural logarithm
`Exp`	${e}^{z}$	Exponential function
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("014c4e"),
    Formula(Where(Less(Abs(Sub(ConstGamma, Sub(Sub(Div(S, I), Div(T, Pow(I, 2))), Log(n)))), Mul(24, Exp(Neg(Mul(8, n))))), Equal(Tuple(S, I, T), Tuple(Sum(Div(Mul(HarmonicNumber(k), Pow(n, Mul(2, k))), Pow(Factorial(k), 2)), Tuple(k, 0, Mul(5, n))), Sum(Div(Pow(n, Mul(2, k)), Pow(Factorial(k), 2)), Tuple(k, 0, Mul(5, n))), Mul(Div(1, Mul(4, n)), Sum(Div(Pow(Factorial(Mul(2, k)), 3), Mul(Mul(Pow(Factorial(k), 4), Pow(8, Mul(2, k))), Pow(Mul(2, n), Mul(2, k)))), Tuple(k, 0, Sub(Mul(2, n), 1)))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))),
    References("R. Brent and F. Johansson. A bound for the error term in the Brent-McMillan algorithm. Mathematics of Computation 2015, 84(295). DOI: 10.1090/S0025-5718-2015-02931-7"))

Numerical value

Limit representations

Special function representations

Integral representations

Approximations

Related topics