DLMF:19.14.E2 (Q6307)

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DLMF:19.14.E2
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    Statements

    Defining formula
    ∫ x 1 d t 1 - t 3 = 3 - 1 / 4 ⁢ F ⁡ ( ϕ , k ) , superscript subscript 𝑥 1 𝑡 1 superscript 𝑡 3 superscript 3 1 4 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle\int_{x}^{1}\frac{\mathrm{d}t}{\sqrt{1-t^{3}}}=3^{% -1/4}F\left(\phi,k\right),}}
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    DLMF id
    DLMF:19.14.E2
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    constraint
    cos ⁡ ϕ = 3 - 1 + x 3 + 1 - x italic-ϕ 3 1 𝑥 3 1 𝑥 {\displaystyle{\displaystyle\cos\phi=\dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x}}}
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    cos ⁡ ϕ = 3 - 1 + x 3 + 1 - x italic-ϕ 3 1 𝑥 3 1 𝑥 {\displaystyle{\displaystyle\cos\phi=\dfrac{\sqrt{3}-1+x}{\sqrt{3}+1-x}}}
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    k 2 = 2 + 3 4 superscript 𝑘 2 2 3 4 {\displaystyle{\displaystyle k^{2}=\dfrac{2+\sqrt{3}}{4}}}
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    Symbols used
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2aadec
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