DLMF:5.12.E6 (Q2151): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2acdec

Revision as of 19:44, 1 January 2020

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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)},}}
    0 references
    DLMF id
    DLMF:5.12.E6
    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.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
     
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