DLMF:5.12.E8 (Q2153)

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DLMF:5.12.E8
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ - ∞ ∞ d t ( w + i ⁢ t ) a ⁢ ( z - i ⁢ t ) b = ( w + z ) 1 - a - b ( a + b - 1 ) ⁢ B ⁡ ( a , b ) , 1 2 𝜋 superscript subscript 𝑡 superscript 𝑤 𝑖 𝑡 𝑎 superscript 𝑧 𝑖 𝑡 𝑏 superscript 𝑤 𝑧 1 𝑎 𝑏 𝑎 𝑏 1 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle{\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{% \mathrm{d}t}{(w+it)^{a}(z-it)^{b}}=\frac{(w+z)^{1-a-b}}{(a+b-1)\mathrm{B}\left% (a,b\right)}},}}
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    DLMF id
    DLMF:5.12.E8
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    constraint
    ℜ ⁡ ( a + b ) > 1 𝑎 𝑏 1 {\displaystyle{\displaystyle\Re(a+b)>1}}
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    ℜ ⁡ w > 0 𝑤 0 {\displaystyle{\displaystyle\Re w>0}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>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.m2agdec
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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.m1agdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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