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Fungrim entry: 6395ee

logG ⁣(z+1)=zlogΓ(z)+z24log(z)2B2 ⁣(z)log(A)0ezxx2(11ex1x12x12)dx\log G\!\left(z + 1\right) = z \log \Gamma(z) + \frac{{z}^{2}}{4} - \frac{\log(z)}{2} B_{2}\!\left(z\right) - \log(A) - \int_{0}^{\infty} \frac{{e}^{-z x}}{{x}^{2}} \left(\frac{1}{1 - {e}^{-x}} - \frac{1}{x} - \frac{1}{2} - \frac{x}{12}\right) \, dx
Assumptions:zC  and  Re(z)>0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
\log G\!\left(z + 1\right) = z \log \Gamma(z) + \frac{{z}^{2}}{4} - \frac{\log(z)}{2} B_{2}\!\left(z\right) - \log(A) - \int_{0}^{\infty} \frac{{e}^{-z x}}{{x}^{2}} \left(\frac{1}{1 - {e}^{-x}} - \frac{1}{x} - \frac{1}{2} - \frac{x}{12}\right) \, dx

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
Fungrim symbol Notation Short description
LogBarnesGlogG(z)\log G(z) Logarithmic Barnes G-function
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
Powab{a}^{b} Power
Loglog(z)\log(z) Natural logarithm
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    Formula(Equal(LogBarnesG(Add(z, 1)), Sub(Sub(Sub(Add(Mul(z, LogGamma(z)), Div(Pow(z, 2), 4)), Mul(Div(Log(z), 2), BernoulliPolynomial(2, z))), Log(ConstGlaisher)), Integral(Mul(Div(Exp(Mul(Neg(z), x)), Pow(x, 2)), Sub(Sub(Sub(Div(1, Sub(1, Exp(Neg(x)))), Div(1, x)), Div(1, 2)), Div(x, 12))), For(x, 0, Infinity))))),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))),

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC