DLMF:19.20.E2 (Q6376)

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DLMF:19.20.E2
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    Statements

    Defining formula
    ∫ 0 1 d t 1 - t 4 = R F ⁡ ( 0 , 1 , 2 ) = ( Γ ⁡ ( 1 4 ) ) 2 4 ⁢ ( 2 ⁢ π ) 1 / 2 = 1.31102 87771 46059 90523 ⁢ … . superscript subscript 0 1 𝑡 1 superscript 𝑡 4 Carlson-integral-RF 0 1 2 superscript Euler-Gamma 1 4 2 4 superscript 2 𝜋 1 2 1.31102 87771 46059 90523 … {\displaystyle{\displaystyle\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}=R_{% F}\left(0,1,2\right)=\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2}}{4(% 2\pi)^{1/2}}=1.31102\;87771\;46059\;90523\;\dots.}}
    0 references
    DLMF id
    DLMF:19.20.E2
    0 references
    Symbols used
    symmetric elliptic integral of first kind
    Defining formula
    R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E1.m2aadec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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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.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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