DLMF:25.11.E8 (Q7682): Difference between revisions

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‎Changed an Item: Add constraint
 
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Property / constraint
 

s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}

s\neq 1
Property / constraint: s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}} / rank
 
Normal rank
Property / constraint
 

0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle 0<a\leq 1}}

0<a\leq 1
Property / constraint: 0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle 0<a\leq 1}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Hurwitz zeta function / rank
 
Normal rank
Property / Symbols used: Hurwitz zeta function / qualifier
 
Defining formula:

ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}

\Hurwitzzeta@{\NVar{s}}{\NVar{a}}
Property / Symbols used: Hurwitz zeta function / qualifier
 
xml-id: C25.S11.E1.m2aidec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1addec
Property / Symbols used
 
Property / Symbols used: Q12148 / rank
 
Normal rank
Property / Symbols used: Q12148 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12148 / qualifier
 
xml-id: C25.S1.XMD3.m1edec
Property / Symbols used
 
Property / Symbols used: Q12161 / rank
 
Normal rank
Property / Symbols used: Q12161 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12161 / qualifier
 
xml-id: C25.S1.XMD6.m1hdec
Property / Symbols used
 
Property / Symbols used: Q12149 / rank
 
Normal rank
Property / Symbols used: Q12149 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q12149 / qualifier
 
xml-id: C25.S1.XMD7.m1gdec

Latest revision as of 13:37, 2 January 2020

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English
DLMF:25.11.E8
No description defined

    Statements

    Defining formula
    ζ ⁡ ( s , 1 2 ⁢ a ) = ζ ⁡ ( s , 1 2 ⁢ a + 1 2 ) + 2 s ⁢ ∑ n = 0 ∞ ( - 1 ) n ( n + a ) s , Hurwitz-zeta 𝑠 1 2 𝑎 Hurwitz-zeta 𝑠 1 2 𝑎 1 2 superscript 2 𝑠 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,% \tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a% )^{s}},}}
    0 references
    DLMF id
    DLMF:25.11.E8
    0 references
    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
    0 references
    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
    0 references
    0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle 0<a\leq 1}}
    0 references
    Symbols used
    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aidec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
    0 references
    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1edec
    0 references
    Q12161
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1hdec
    0 references
    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1gdec
    0 references
     
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