DLMF:19.4.E7 (Q6127): Difference between revisions

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

Π ⁡ ( ϕ , α 2 , k ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}% \right)}}

\incellintPik@{\NVar{\phi}}{\NVar{\alpha}^{2}}{\NVar{k}}
Property / Symbols used: Legendre’s incomplete elliptic integral of the third kind / qualifier
 
xml-id: C19.S2.E7.m2adec

Revision as of 12:37, 2 January 2020

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DLMF:19.4.E7
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    Statements

    test
    ∂ ⁡ Π ⁡ ( ϕ , α 2 , k ) ∂ ⁡ k = k k ′ 2 ⁢ ( k 2 - α 2 ) ⁢ ( E ⁡ ( ϕ , k ) - k ′ 2 ⁢ Π ⁡ ( ϕ , α 2 , k ) - k 2 ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ 1 - k 2 ⁢ sin 2 ⁡ ϕ ) . partial-derivative elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 𝑘 𝑘 superscript superscript 𝑘 ′ 2 superscript 𝑘 2 superscript 𝛼 2 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript superscript 𝑘 ′ 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript 𝑘 2 italic-ϕ italic-ϕ 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{% \partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k% \right)-{k^{\prime}}^{2}\Pi\left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi% \cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right).}}
    0 references
    DLMF id
    DLMF:19.4.E7
    0 references
    Symbols used
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    0 references
    Legendre’s incomplete elliptic integral of the second kind
    test
    E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E5.m2abdec
    0 references
    Legendre’s incomplete elliptic integral of the third kind
    test
    Π ⁡ ( ϕ , α 2 , k ) elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}% \right)}}
    xml-id
    C19.S2.E7.m2adec
    0 references
     
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