DLMF:1.14.E37 (Q529)

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DLMF:1.14.E37
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    Statements

    test
    ∫ 0 ∞ f ⁢ ( x ) ⁢ g ⁢ ( x ) ⁢ d x = 1 2 ⁢ π ⁢ i ⁢ ∫ σ - i ⁢ ∞ σ + i ⁢ ∞ ℳ ⁡ f ⁡ ( 1 - s ) ⁢ ℳ ⁡ g ⁡ ( s ) ⁢ d s . subscript superscript 0 𝑓 𝑥 𝑔 𝑥 𝑥 1 2 𝜋 𝑖 subscript superscript 𝜎 𝑖 𝜎 𝑖 Mellin-transform 𝑓 1 𝑠 Mellin-transform 𝑔 𝑠 𝑠 {\displaystyle{\displaystyle\int^{\infty}_{0}f(x)g(x)\mathrm{d}x=\frac{1}{2\pi i% }\*\int^{\sigma+i\infty}_{\sigma-i\infty}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu% \left(1-s\right)\mathscr{M}\mskip-3.0mu g\mskip 3.0mu \left(s\right)\mathrm{d% }s.}}
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    DLMF id
    DLMF:1.14.E37
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    Symbols used
    Mellin transform
    test
    ℳ ⁡ ( f ) ⁡ ( s ) Mellin-transform 𝑓 𝑠 {\displaystyle{\displaystyle\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}% \right)}}
    xml-id
    C1.S14.E32.m2aedec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1azdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aidec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aydec
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    Q10894
    test
    σ ∈ ( a , b ) 𝜎 𝑎 𝑏 {\displaystyle{\displaystyle\sigma\in(a,b)}}
    xml-id
    C1.S14.XMD11.m1edec
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