DLMF:25.4.E5 (Q7612): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12155 / rank
 
Normal rank
Property / Symbols used: Q12155 / qualifier
 
test:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q12155 / qualifier
 
xml-id: C25.S4.XMD1.m1dec

Revision as of 12:23, 2 January 2020

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DLMF:25.4.E5
No description defined

    Statements

    test
    ( - 1 ) k ⁢ ζ ( k ) ⁡ ( 1 - s ) = 2 ( 2 ⁢ π ) s ⁢ ∑ m = 0 k ∑ r = 0 m ( k m ) ⁢ ( m r ) ⁢ ( ℜ ⁡ ( c k - m ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s ) + ℑ ⁡ ( c k - m ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ s ) ) ⁢ Γ ( r ) ⁡ ( s ) ⁢ ζ ( m - r ) ⁡ ( s ) , superscript 1 𝑘 Riemann-zeta 𝑘 1 𝑠 2 superscript 2 𝜋 𝑠 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 1 2 𝜋 𝑠 superscript 𝑐 𝑘 𝑚 1 2 𝜋 𝑠 Euler-Gamma 𝑟 𝑠 Riemann-zeta 𝑚 𝑟 𝑠 {\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-s\right)=\frac{2}{(2% \pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{% )}{0.0pt}{}{m}{r}\left(\Re(c^{k-m})\cos\left(\tfrac{1}{2}\pi s\right)+\Im(c^{k% -m})\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}\left(s\right){% \zeta^{(m-r)}}\left(s\right),}}
    0 references
    DLMF id
    DLMF:25.4.E5
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Riemann zeta function
    test
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2acdec
    0 references
    Q10754
    test
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Q10901
    test
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2adec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q12150
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1dec
    0 references
    Q12151
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C25.S1.XMD2.m1dec
    0 references
    Q12149
    test
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1ddec
    0 references
    Q12155
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C25.S4.XMD1.m1dec
    0 references
     
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