DLMF:5.12.E10 (Q2155): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11283 / rank
 
Normal rank
Property / Symbols used: Q11283 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11283 / qualifier
 
xml-id: C5.S1.XMD9.m1idec

Latest revision as of 01:03, 2 January 2020

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DLMF:5.12.E10
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ i ⁢ ∫ 0 ( 1 + ) t a - 1 ⁢ ( t - 1 ) b - 1 ⁢ d t = sin ⁡ ( π ⁢ b ) π ⁢ B ⁡ ( a , b ) , 1 2 𝜋 𝑖 superscript subscript 0 limit-from 1 superscript 𝑡 𝑎 1 superscript 𝑡 1 𝑏 1 𝑡 𝜋 𝑏 𝜋 Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle{\frac{1}{2\pi i}\int_{0}^{(1+)}t^{a-1}(t-1)^{b-1}% \mathrm{d}t=\frac{\sin\left(\pi b\right)}{\pi}\mathrm{B}\left(a,b\right)},}}
    0 references
    DLMF id
    DLMF:5.12.E10
    0 references
    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    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.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1idec
    0 references
    Q11283
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C5.S1.XMD9.m1idec
    0 references
     
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