2.5.E7 (Q753)

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2.5.E7
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    Statements

    DLMF defining formula
    E ⁢ ( x ) = 1 2 ⁢ π ⁢ i ⁢ ∫ d - i ⁢ ∞ d + i ⁢ ∞ x - z ⁢ ℳ ⁡ f ⁡ ( 1 - z ) ⁢ ℳ ⁡ h ⁡ ( z ) ⁢ d z . 𝐸 𝑥 1 2 𝜋 𝑖 superscript subscript 𝑑 𝑖 𝑑 𝑖 superscript 𝑥 𝑧 Mellin-transform 𝑓 1 𝑧 Mellin-transform ℎ 𝑧 𝑧 {\displaystyle{\displaystyle E(x)=\frac{1}{2\pi i}\int_{d-i\infty}^{d+i\infty}% x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)\mathscr{M}\mskip-% 3.0mu h\mskip 3.0mu \left(z\right)\mathrm{d}z.}}
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    Symbols used
    Mellin transform
    DLMF 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.m2aedec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    locally integrable function
    DLMF defining formula
    f ⁢ ( x ) 𝑓 𝑥 {\displaystyle{\displaystyle f(x)}}
    xml-id
    C2.S5.XMD1.m1fdec
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    function
    DLMF defining formula
    h ⁢ ( x ) ℎ 𝑥 {\displaystyle{\displaystyle h(x)}}
    xml-id
    C2.S5.XMD6.m1ddec
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    point
    DLMF defining formula
    d 𝑑 {\displaystyle{\displaystyle d}}
    xml-id
    C2.S5.XMD7.m1adec
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    function
    DLMF defining formula
    E ⁢ ( x ) 𝐸 𝑥 {\displaystyle{\displaystyle E(x)}}
    xml-id
    C2.S5.XMD8.m1adec
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    Identifiers

     
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