DLMF:5.12.E12 (Q2157)

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DLMF:5.12.E12
No description defined

    Statements

    Defining formula
    ∫ P ( 1 + , 0 + , 1 - , 0 - ) t a - 1 ⁢ ( 1 - t ) b - 1 ⁢ d t = - 4 ⁢ e π ⁢ i ⁢ ( a + b ) ⁢ sin ⁡ ( π ⁢ a ) ⁢ sin ⁡ ( π ⁢ b ) ⁢ B ⁡ ( a , b ) , superscript subscript 𝑃 limit-from 1 limit-from 0 limit-from 1 limit-from 0 superscript 𝑡 𝑎 1 superscript 1 𝑡 𝑏 1 𝑡 4 superscript 𝑒 𝜋 𝑖 𝑎 𝑏 𝜋 𝑎 𝜋 𝑏 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\mathrm{% d}t=-4e^{\pi i(a+b)}\sin\left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}% \left(a,b\right),}}
    0 references
    DLMF id
    DLMF:5.12.E12
    0 references
    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.m2akdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
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    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1kdec
    0 references
    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1kdec
    0 references
    Q11291
    Defining formula
    P 𝑃 {\displaystyle{\displaystyle P}}
    xml-id
    C5.S12.XMD1.m1dec
    0 references
     
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