DLMF:2.5.E40 (Q816)

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DLMF:2.5.E40
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    Statements

    Defining formula
    I j ⁢ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ p j - i ⁢ ∞ p j + i ⁢ ∞ x - z ⁢ Γ ⁡ ( 1 - z ) ⁢ ℳ ⁡ h j ⁡ ( z ) ⁢ d z , subscript 𝐼 𝑗 𝑥 1 2 𝜋 𝑖 superscript subscript subscript 𝑝 𝑗 𝑖 subscript 𝑝 𝑗 𝑖 superscript 𝑥 𝑧 Euler-Gamma 1 𝑧 Mellin-transform subscript ℎ 𝑗 𝑧 𝑧 {\displaystyle{\displaystyle I_{j}(x)=\frac{1}{2\pi i}\int_{p_{j}-i\infty}^{p_% {j}+i\infty}x^{-z}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0mu h_{j}\mskip 3.% 0mu \left(z\right)\mathrm{d}z,}}
    0 references
    DLMF id
    DLMF:2.5.E40
    0 references
    constraint
    j = 1 , 2 𝑗 1 2 {\displaystyle{\displaystyle j=1,2}}
    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.m2acdec
    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.m2andec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
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    Q11005
    Defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD16.m1ldec
    0 references
    Q11014
    Defining formula
    I j ⁢ ( x ) subscript 𝐼 𝑗 𝑥 {\displaystyle{\displaystyle I_{j}(x)}}
    xml-id
    C2.S5.XMD25.m1bdec
    0 references
    Q11015
    Defining formula
    p j subscript 𝑝 𝑗 {\displaystyle{\displaystyle p_{j}}}
    xml-id
    C2.S5.XMD26.m1dec
    0 references
     
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