DLMF:19.20.E22 (Q6413)

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DLMF:19.20.E22
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    Statements

    test
    ∫ 0 1 t 2 ⁢ d t 1 - t 4 = 1 3 ⁢ R D ⁡ ( 0 , 2 , 1 ) = ( Γ ⁡ ( 3 4 ) ) 2 ( 2 ⁢ π ) 1 / 2 = 0.59907 01173 67796 10371 ⁢ … . superscript subscript 0 1 superscript 𝑡 2 𝑡 1 superscript 𝑡 4 1 3 Carlson-integral-RD 0 2 1 superscript Euler-Gamma 3 4 2 superscript 2 𝜋 1 2 0.59907 01173 67796 10371 … {\displaystyle{\displaystyle\int_{0}^{1}\frac{t^{2}\mathrm{d}t}{\sqrt{1-t^{4}}% }=\tfrac{1}{3}R_{D}\left(0,2,1\right)=\frac{\left(\Gamma\left(\frac{3}{4}% \right)\right)^{2}}{(2\pi)^{1/2}}=0.59907\;01173\;67796\;10371\dots.}}
    0 references
    DLMF id
    DLMF:19.20.E22
    0 references
    Symbols used
    elliptic integral symmetric in only two variables
    test
    R D ⁡ ( x , y , z ) Carlson-integral-RD 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E5.m2agdec
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    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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