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Fungrim entry: d9c818

ψ ⁣(z)=log(z)12z20t(t2+z2)(e2πt1)dt\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - 2 \int_{0}^{\infty} \frac{t}{\left({t}^{2} + {z}^{2}\right) \left({e}^{2 \pi t} - 1\right)} \, dt
Assumptions:zCandRe(z)>0z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) > 0
TeX:
\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - 2 \int_{0}^{\infty} \frac{t}{\left({t}^{2} + {z}^{2}\right) \left({e}^{2 \pi t} - 1\right)} \, dt

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(z) > 0
Definitions:
Fungrim symbol Notation Short description
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Loglog(z)\log(z) Natural logarithm
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
Piπ\pi The constant pi (3.14...)
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("d9c818"),
    Formula(Equal(DigammaFunction(z), Sub(Sub(Log(z), Div(1, Mul(2, z))), Mul(2, Integral(Div(t, Mul(Add(Pow(t, 2), Pow(z, 2)), Sub(Exp(Mul(Mul(2, Pi), t)), 1))), For(t, 0, Infinity)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))))

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2020-01-31 18:09:28.494564 UTC