DLMF:5.12.E6 (Q2151)

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DLMF:5.12.E6
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    Statements

    Defining formula
    ∫ 0 π ( sin ⁡ t ) a - 1 ⁢ e i ⁢ b ⁢ t ⁢ d t = π 2 a - 1 ⁢ e i ⁢ π ⁢ b / 2 a ⁢ B ⁡ ( 1 2 ⁢ ( a + b + 1 ) , 1 2 ⁢ ( a - b + 1 ) ) , superscript subscript 0 𝜋 superscript 𝑡 𝑎 1 superscript 𝑒 𝑖 𝑏 𝑡 𝑡 𝜋 superscript 2 𝑎 1 superscript 𝑒 𝑖 𝜋 𝑏 2 𝑎 Euler-Beta 1 2 𝑎 𝑏 1 1 2 𝑎 𝑏 1 {\displaystyle{\displaystyle\int_{0}^{\pi}(\sin t)^{a-1}e^{ibt}\mathrm{d}t=% \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\mathrm{B}\left(\frac{1}{2}(a+b+1),% \frac{1}{2}(a-b+1)\right)},}}
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    DLMF id
    DLMF:5.12.E6
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>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.m2aedec
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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.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1edec
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    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1edec
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