DLMF:2.5.E47 (Q823)

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DLMF:2.5.E47
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    Statements

    Defining formula
    res z = 1 ⁡ [ - ζ z - 1 ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 2 ⁡ ( z ) ] = ( - ln ⁡ ζ - γ ) - ℳ ⁡ h 1 ⁡ ( 1 ) , subscript residue 𝑧 1 delimited-[] superscript 𝜁 𝑧 1 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 2 𝑧 𝜁 Mellin-transform subscript ℎ 1 1 {\displaystyle{\displaystyle\Residue_{z=1}\left[-\zeta^{z-1}\Gamma\left(1-z% \right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\right]=\left(-% \ln\zeta-\gamma\right)-\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right% ),}}
    0 references
    DLMF id
    DLMF:2.5.E47
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
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    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
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    Mellin transform
    Defining formula
    ℳ ⁡ ( f ) ⁡ ( s ) Mellin-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E32.m2asdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aedec
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    Q10960
    Defining formula
    res residue {\displaystyle{\displaystyle\Residue}}
    xml-id
    C1.S10.SS3.p5.m4afdec
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    Q11005
    Defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD16.m1rdec
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