DLMF:5.13.E2 (Q2159)

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DLMF:5.13.E2
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    Statements

    test
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ | Γ ⁡ ( a + i ⁢ t ) | 2 ⁢ e ( 2 ⁢ b - π ) ⁢ t ⁢ d t = Γ ⁡ ( 2 ⁢ a ) ( 2 ⁢ sin ⁡ b ) 2 ⁢ a , 1 2 𝜋 superscript subscript superscript Euler-Gamma 𝑎 𝑖 𝑡 2 superscript 𝑒 2 𝑏 𝜋 𝑡 𝑡 Euler-Gamma 2 𝑎 superscript 2 𝑏 2 𝑎 {\displaystyle{\displaystyle{\frac{1}{2\pi}\int_{-\infty}^{\infty}|\Gamma\left% (a+it\right)|^{2}e^{(2b-\pi)t}\mathrm{d}t=\frac{\Gamma\left(2a\right)}{(2\sin b% )^{2a}}},}}
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    DLMF id
    DLMF:5.13.E2
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    constraint
    0 < b < π 0 𝑏 𝜋 {\displaystyle{\displaystyle 0<b<\pi}}
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    Symbols used
    gamma function
    test
    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
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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    Q11275
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1adec
    0 references
     
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