DLMF:19.14.E1 (Q6306)

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DLMF:19.14.E1
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    Statements

    Defining formula
    ∫ 1 x d t t 3 - 1 = 3 - 1 / 4 ⁒ F ⁑ ( Ο• , k ) , superscript subscript 1 π‘₯ 𝑑 superscript 𝑑 3 1 superscript 3 1 4 elliptic-integral-first-kind-F italic-Ο• π‘˜ {\displaystyle{\displaystyle\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-1}}=3^{% -1/4}F\left(\phi,k\right),}}
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    DLMF id
    DLMF:19.14.E1
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    constraint
    cos ⁑ Ο• = 3 + 1 - x 3 - 1 + x italic-Ο• 3 1 π‘₯ 3 1 π‘₯ {\displaystyle{\displaystyle\cos\phi=\dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x}}}
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    cos ⁑ Ο• = 3 + 1 - x 3 - 1 + x italic-Ο• 3 1 π‘₯ 3 1 π‘₯ {\displaystyle{\displaystyle\cos\phi=\dfrac{\sqrt{3}+1-x}{\sqrt{3}-1+x}}}
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    k 2 = 2 - 3 4 superscript π‘˜ 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.m2adec
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    Q10770
    Defining formula
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    F ⁑ ( Ο• , k ) elliptic-integral-first-kind-F italic-Ο• π‘˜ {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q11807
    Defining formula
    Ο• italic-Ο• {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1dec
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    Q11808
    Defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1dec
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