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Fungrim entry: 4f5575

ψ ⁣(z)=log(z)12z0ezt(121t+1et1)dt\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{{e}^{t} - 1}\right) \, dt
Assumptions:zC  and  Re(z)>0z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{{e}^{t} - 1}\right) \, dt

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > 0
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
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(DigammaFunction(z), Sub(Sub(Log(z), Div(1, Mul(2, z))), Integral(Mul(Exp(Neg(Mul(z, t))), Parentheses(Add(Sub(Div(1, 2), Div(1, t)), Div(1, Sub(Exp(t), 1))))), For(t, 0, Infinity))))),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))))

Topics using this entry

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