DLMF:25.11.E5 (Q7679): Difference between revisions

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Property / constraint
 

ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}

\realpart@@{s}>0
Property / constraint: ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}} / rank
 
Normal rank
Property / constraint
 

a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}

a>0
Property / constraint: a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}} / rank
 
Normal rank
Property / constraint
 

N = 0 , 1 , 2 , 3 , … 𝑁 0 1 2 3 … {\displaystyle{\displaystyle N=0,1,2,3,\dots}}

N=0,1,2,3,\dots
Property / constraint: N = 0 , 1 , 2 , 3 , … 𝑁 0 1 2 3 … {\displaystyle{\displaystyle N=0,1,2,3,\dots}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Hurwitz zeta function / rank
 
Normal rank
Property / Symbols used: Hurwitz zeta function / qualifier
 
test:

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.m2afdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: Q11169 / rank
 
Normal rank
Property / Symbols used: Q11169 / qualifier
 
test:

⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}

\floor{\NVar{x}}
Property / Symbols used: Q11169 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r17.m4adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aadec
Property / Symbols used
 
Property / Symbols used: Q12148 / rank
 
Normal rank
Property / Symbols used: Q12148 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12148 / qualifier
 
xml-id: C25.S1.XMD3.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12152 / rank
 
Normal rank
Property / Symbols used: Q12152 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12152 / qualifier
 
xml-id: C25.S1.XMD5.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12161 / rank
 
Normal rank
Property / Symbols used: Q12161 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12161 / qualifier
 
xml-id: C25.S1.XMD6.m1edec
Property / Symbols used
 
Property / Symbols used: Q12149 / rank
 
Normal rank
Property / Symbols used: Q12149 / qualifier
 
test:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q12149 / qualifier
 
xml-id: C25.S1.XMD7.m1ddec

Latest revision as of 12:36, 2 January 2020

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DLMF:25.11.E5
No description defined

    Statements

    test
    ζ ⁡ ( s , a ) = ∑ n = 0 N 1 ( n + a ) s + ( N + a ) 1 - s s - 1 - s ⁢ ∫ N ∞ x - ⌊ x ⌋ ( x + a ) s + 1 ⁢ d x , Hurwitz-zeta 𝑠 𝑎 superscript subscript 𝑛 0 𝑁 1 superscript 𝑛 𝑎 𝑠 superscript 𝑁 𝑎 1 𝑠 𝑠 1 𝑠 superscript subscript 𝑁 𝑥 𝑥 superscript 𝑥 𝑎 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)% ^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right% \rfloor}{(x+a)^{s+1}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.11.E5
    0 references
    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
    0 references
    a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}}
    0 references
    N = 0 , 1 , 2 , 3 , … 𝑁 0 1 2 3 … {\displaystyle{\displaystyle N=0,1,2,3,\dots}}
    0 references
    Symbols used
    Hurwitz zeta function
    test
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2afdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q11169
    test
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q12148
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1cdec
    0 references
    Q12152
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1bdec
    0 references
    Q12161
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1edec
    0 references
    Q12149
    test
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1ddec
    0 references
     
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