DLMF:5.12.E9 (Q2154)

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DLMF:5.12.E9
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ c - ∞ ⁢ i c + ∞ ⁢ i t - a ⁢ ( 1 - t ) - 1 - b ⁢ d t = 1 b ⁢ B ⁡ ( a , b ) , 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝑡 𝑎 superscript 1 𝑡 1 𝑏 𝑡 1 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle{\frac{1}{2\pi i}\int_{c-\infty i}^{c+\infty i}t^{% -a}(1-t)^{-1-b}\mathrm{d}t=\frac{1}{b\mathrm{B}\left(a,b\right)}},}}
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    DLMF id
    DLMF:5.12.E9
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    constraint
    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
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    0 < c < 1 0 𝑐 1 {\displaystyle{\displaystyle 0<c<1}}
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    ℜ ⁡ ( a + b ) > 0 𝑎 𝑏 0 {\displaystyle{\displaystyle\Re(a+b)>0}}
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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.m2ahdec
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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.m2aedec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1addec
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    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1hdec
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    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1hdec
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