DLMF:2.10.E22 (Q1009)

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DLMF:2.10.E22
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    Statements

    Defining formula
    ∑ j = 0 n - 1 x j ( j ! ) 3 = ∫ - 1 / 2 n - ( 1 / 2 ) x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t - ∫ 𝒞 1 x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t e - 2 ⁢ π ⁢ i ⁢ t - 1 + ∫ 𝒞 2 x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t e 2 ⁢ π ⁢ i ⁢ t - 1 , superscript subscript 𝑗 0 𝑛 1 superscript 𝑥 𝑗 superscript 𝑗 3 superscript subscript 1 2 𝑛 1 2 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 subscript subscript 𝒞 1 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 superscript 𝑒 2 𝜋 𝑖 𝑡 1 subscript subscript 𝒞 2 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 superscript 𝑒 2 𝜋 𝑖 𝑡 1 {\displaystyle{\displaystyle\sum_{j=0}^{n-1}\frac{x^{j}}{(j!)^{3}}=\int_{-1/2}% ^{n-(1/2)}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\mathrm{d}t-\int_{\mathscr% {C}_{1}}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\frac{\mathrm{d}t}{e^{-2\pi it% }-1}+\int_{\mathscr{C}_{2}}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\frac{% \mathrm{d}t}{e^{2\pi it}-1},}}
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    DLMF id
    DLMF:2.10.E22
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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