DLMF:25.11.E7 (Q7681): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q12150 / rank
 
Normal rank
Property / Symbols used: Q12150 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12150 / qualifier
 
xml-id: C25.S1.XMD1.m1dec

Revision as of 13:37, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:25.11.E7
No description defined

    Statements

    Defining formula
    ζ ⁡ ( s , a ) = 1 a s + 1 ( 1 + a ) s ⁢ ( 1 2 + 1 + a s - 1 ) + ∑ k = 1 n ( s + 2 ⁢ k - 2 2 ⁢ k - 1 ) ⁢ B 2 ⁢ k 2 ⁢ k ⁢ 1 ( 1 + a ) s + 2 ⁢ k - 1 - ( s + 2 ⁢ n 2 ⁢ n + 1 ) ⁢ ∫ 1 ∞ B ~ 2 ⁢ n + 1 ⁡ ( x ) ( x + a ) s + 2 ⁢ n + 1 ⁢ d x , Hurwitz-zeta 𝑠 𝑎 1 superscript 𝑎 𝑠 1 superscript 1 𝑎 𝑠 1 2 1 𝑎 𝑠 1 superscript subscript 𝑘 1 𝑛 binomial 𝑠 2 𝑘 2 2 𝑘 1 Bernoulli-number-B 2 𝑘 2 𝑘 1 superscript 1 𝑎 𝑠 2 𝑘 1 binomial 𝑠 2 𝑛 2 𝑛 1 superscript subscript 1 periodic-Bernoulli-polynomial-B 2 𝑛 1 𝑥 superscript 𝑥 𝑎 𝑠 2 𝑛 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+% a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0% .0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{% 0.0pt}{}{s+2n}{2n+1}\int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}% {(x+a)^{s+2n+1}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.11.E7
    0 references
    constraint
    ℜ ⁡ s > - 2 ⁢ n 𝑠 2 𝑛 {\displaystyle{\displaystyle\Re s>-2n}}
    0 references
    n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle n=1,2,3,\dots}}
    0 references
    ℜ ⁡ s > - 2 ⁢ n 𝑠 2 𝑛 {\displaystyle{\displaystyle\Re s>-2n}}
    0 references
    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1aadec
    0 references
    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.m2ahdec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q11110
    Defining formula
    B ~ n ⁡ ( x ) periodic-Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS3.m1aadec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q7681&oldid=38314"