Fungrim entry: bed7ee

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

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

TeX:

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

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \left(a \notin \{0, -1, \ldots\} \;\mathbin{\operatorname{or}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{or}}\; s = 0\right) \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

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

Source code for this entry:

Entry(ID("bed7ee"),
    Formula(Equal(HurwitzZeta(s, Add(a, N)), Sub(HurwitzZeta(s, a), Sum(Div(1, Pow(Add(n, a), s)), For(n, 0, Sub(N, 1)))))),
    Variables(s, a, N),
    Assumptions(And(Element(s, CC), Element(a, CC), NotEqual(s, 1), Or(NotElement(a, ZZLessEqual(0)), Less(Re(s), 0), Equal(s, 0)), Element(N, ZZGreaterEqual(1)))))

Topics using this entry

Hurwitz zeta function