DLMF:5.12.E11 (Q2156)

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DLMF:5.12.E11
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    Statements

    Defining formula
    1 e 2 ⁢ π ⁢ i ⁢ a - 1 ⁢ ∫ ∞ ( 0 + ) t a - 1 ⁢ ( 1 + t ) - a - b ⁢ d t = B ⁡ ( a , b ) , 1 superscript 𝑒 2 𝜋 𝑖 𝑎 1 superscript subscript limit-from 0 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑎 𝑏 𝑡 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}% (1+t)^{-a-b}\mathrm{d}t=\mathrm{B}\left(a,b\right),}}
    0 references
    DLMF id
    DLMF:5.12.E11
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    Symbols used
    beta function
    Defining formula
    B ⁡ ( a , b ) Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C5.S12.E1.m2ajdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
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