DLMF:25.11.E28 (Q7702): Difference between revisions

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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.m1jdec
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.m1wdec
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.m1tdec

Latest revision as of 12:42, 2 January 2020

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English
DLMF:25.11.E28
No description defined

    Statements

    test
    ζ ⁡ ( s , a ) = 1 2 ⁢ a - s + a 1 - s s - 1 + ∑ k = 1 n B 2 ⁢ k ( 2 ⁢ k ) ! ⁢ ( s ) 2 ⁢ k - 1 ⁢ a 1 - s - 2 ⁢ k + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 - ∑ k = 1 n B 2 ⁢ k ( 2 ⁢ k ) ! ⁢ x 2 ⁢ k - 1 ) ⁢ x s - 1 ⁢ e - a ⁢ x ⁢ d x , Hurwitz-zeta 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 superscript subscript 𝑘 1 𝑛 Bernoulli-number-B 2 𝑘 2 𝑘 Pochhammer 𝑠 2 𝑘 1 superscript 𝑎 1 𝑠 2 𝑘 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript subscript 𝑘 1 𝑛 Bernoulli-number-B 2 𝑘 2 𝑘 superscript 𝑥 2 𝑘 1 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1% -s}}{s-1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1% }{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-% ax}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.11.E28
    0 references
    constraint
    ℜ ⁡ s > - ( 2 ⁢ n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}
    0 references
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    ℜ ⁡ s > - ( 2 ⁢ n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle\Re s>-(2n+1)}}
    0 references
    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
    0 references
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    Symbols used
    Q11109
    test
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1agdec
    0 references
    order not exceeding
    test
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    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.m2aacdec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aldec
    0 references
    Q12150
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1cdec
    0 references
    Q12148
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1ndec
    0 references
    Q12152
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1jdec
    0 references
    Q12161
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1wdec
    0 references
    Q12149
    test
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1tdec
    0 references
     
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