Fungrim entry: 693e0e

\psi\!\left(z\right) = \lim_{s \to 1} \left[\frac{1}{s - 1} - \zeta\!\left(s, z\right)\right]

Assumptions:

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

TeX:

\psi\!\left(z\right) = \lim_{s \to 1} \left[\frac{1}{s - 1} - \zeta\!\left(s, z\right)\right]

z \in \mathbb{C} \setminus \{0, -1, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`DigammaFunction`	$\psi\!\left(z\right)$	Digamma function
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("693e0e"),
    Formula(Equal(DigammaFunction(z), ComplexLimit(Brackets(Sub(Div(1, Sub(s, 1)), HurwitzZeta(s, z))), For(s, 1)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, ZZLessEqual(0)))))

Topics using this entry

Hurwitz zeta function

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