DLMF:19.14.E1 (Q6306): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Legendre’s incomplete elliptic integral of the first kind / rank
 
Normal rank
Property / Symbols used: Legendre’s incomplete elliptic integral of the first kind / qualifier
 
test:

F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}

\incellintFk@{\NVar{\phi}}{\NVar{k}}
Property / Symbols used: Legendre’s incomplete elliptic integral of the first kind / qualifier
 
xml-id: C19.S2.E4.m2adec

Revision as of 12:50, 2 January 2020

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DLMF:19.14.E1
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    Statements

    test
    ∫ 1 x d t t 3 - 1 = 3 - 1 / 4 ⁢ F ⁡ ( ϕ , k ) , superscript subscript 1 𝑥 𝑡 superscript 𝑡 3 1 superscript 3 1 4 elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle\int_{1}^{x}\frac{\mathrm{d}t}{\sqrt{t^{3}-1}}=3^{% -1/4}F\left(\phi,k\right),}}
    0 references
    DLMF id
    DLMF:19.14.E1
    0 references
    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}}}
    0 references
    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}}}
    0 references
    k 2 = 2 - 3 4 superscript 𝑘 2 2 3 4 {\displaystyle{\displaystyle k^{2}=\dfrac{2-\sqrt{3}}{4}}}
    0 references
    Symbols used
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Legendre’s incomplete elliptic integral of the first kind
    test
    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.m2adec
    0 references
     
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