DLMF:2.5.E42 (Q818)

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DLMF:2.5.E42
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    Statements

    Defining formula
    I 2 ⁢ ( x ) = ∑ ℜ ⁡ β 0 ≤ ℜ ⁡ z ≤ 1 res ⁡ [ - x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 2 ⁡ ( z ) ] + 1 2 ⁢ π ⁢ i ⁢ ∫ ρ - i ⁢ ∞ ρ + i ⁢ ∞ x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 2 ⁡ ( z ) ⁢ d z . subscript 𝐼 2 𝑥 subscript subscript 𝛽 0 𝑧 1 residue delimited-[] superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 2 𝑧 1 2 𝜋 𝑖 superscript subscript 𝜌 𝑖 𝜌 𝑖 superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 2 𝑧 𝑧 {\displaystyle{\displaystyle I_{2}(x)=\sum_{\Re\beta_{0}\leq\Re z\leq 1}% \Residue\left[-x^{-z}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3% .0mu \left(z\right)\right]+\frac{1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x% ^{-z}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z% \right)\mathrm{d}z.}}
    0 references
    DLMF id
    DLMF:2.5.E42
    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.m2aedec
    0 references
    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.m2apdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ajdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    Q10960
    Defining formula
    res residue {\displaystyle{\displaystyle\Residue}}
    xml-id
    C1.S10.SS3.p5.m4acdec
    0 references
    Q11005
    Defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD16.m1ndec
    0 references
    Q11014
    Defining formula
    I j ⁢ ( x ) subscript 𝐼 𝑗 𝑥 {\displaystyle{\displaystyle I_{j}(x)}}
    xml-id
    C2.S5.XMD25.m1ddec
    0 references
    Q11016
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C2.S5.XMD27.m1adec
    0 references
     
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