Fungrim entry: d9c818

\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:

z \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
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`Log`	$\log(z)$	Natural logarithm
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	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))))

Topics using this entry

Digamma function