DLMF:2.5.E41 (Q817)

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DLMF:2.5.E41
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    Statements

    Defining formula
    I 1 ⁢ ( x ) = ℳ ⁡ h 1 ⁡ ( 1 ) ⁢ x - 1 + 1 2 ⁢ π ⁢ i ⁢ ∫ ρ - i ⁢ ∞ ρ + i ⁢ ∞ x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h 1 ⁡ ( z ) ⁢ d z , subscript 𝐼 1 𝑥 Mellin-transform subscript ℎ 1 1 superscript 𝑥 1 1 2 𝜋 𝑖 superscript subscript 𝜌 𝑖 𝜌 𝑖 superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 1 𝑧 𝑧 {\displaystyle{\displaystyle I_{1}(x)=\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0% mu \left(1\right)x^{-1}+\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_{1}\mskip 3.0mu \left(z% \right)\mathrm{d}z,}}
    0 references
    DLMF id
    DLMF:2.5.E41
    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.m2addec
    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.m2aodec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aidec
    0 references
     
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