DLMF:1.14.E36 (Q528)

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DLMF:1.14.E36
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    Statements

    test
    ∫ 0 ∞ f ⁢ ( x ) ⁢ g ⁢ ( y ⁢ x ) ⁢ d x = 1 2 ⁢ π ⁢ i ⁢ ∫ σ - i ⁢ ∞ σ + i ⁢ ∞ y - s ⁢ ℳ ⁡ f ⁡ ( 1 - s ) ⁢ ℳ ⁡ g ⁡ ( s ) ⁢ d s , subscript superscript 0 𝑓 𝑥 𝑔 𝑦 𝑥 𝑥 1 2 𝜋 𝑖 subscript superscript 𝜎 𝑖 𝜎 𝑖 superscript 𝑦 𝑠 Mellin-transform 𝑓 1 𝑠 Mellin-transform 𝑔 𝑠 𝑠 {\displaystyle{\displaystyle\int^{\infty}_{0}f(x)g(yx)\mathrm{d}x=\frac{1}{2% \pi i}\*\int^{\sigma+i\infty}_{\sigma-i\infty}y^{-s}\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,}}
    0 references
    DLMF id
    DLMF:1.14.E36
    0 references
    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.m2addec
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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.m2amdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aydec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3axdec
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