Fungrim entry: 77e507

\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Log`	$\log(z)$	Natural logarithm
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("77e507"),
    Formula(Equal(HurwitzZeta(s, a, r), Mul(Pow(-1, r), Sum(Div(Pow(Log(Add(n, a)), r), Pow(Add(n, a), s)), For(n, 0, Infinity))))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), Greater(Re(s), 1), Element(a, SetMinus(CC, ZZLessEqual(0))), Element(r, ZZGreaterEqual(0)))))

Topics using this entry

Hurwitz zeta function