DLMF:19.7.E9 (Q6204): 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.m2addec

Revision as of 12:41, 2 January 2020

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DLMF:19.7.E9
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    Statements

    test
    Π ⁡ ( ϕ , α 2 , k ) + Π ⁡ ( ϕ , ω 2 , k ) ⁢ ( k 2 - α 2 ) ⁢ Π ⁡ ( ϕ , α 2 , k ) + ( k 2 - ω 2 ) ⁢ Π ⁡ ( ϕ , ω 2 , k ) = k 2 ⁢ F ⁡ ( ϕ , k ) - α 2 ⁢ ω 2 ⁢ c - 1 ⁢ R C ⁡ ( c ⁢ ( c - k 2 ) , ( c - α 2 ) ⁢ ( c - ω 2 ) ) , elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝜔 2 𝑘 superscript 𝑘 2 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript 𝑘 2 superscript 𝜔 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝜔 2 𝑘 superscript 𝑘 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 superscript 𝛼 2 superscript 𝜔 2 𝑐 1 Carlson-integral-RC 𝑐 𝑐 superscript 𝑘 2 𝑐 superscript 𝛼 2 𝑐 superscript 𝜔 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,% \omega^{2},k\right)(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(k^{2}-% \omega^{2})\Pi\left(\phi,\omega^{2},k\right)=k^{2}F\left(\phi,k\right)-\alpha^% {2}\omega^{2}\sqrt{c-1}R_{C}\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})% \right),}}
    0 references
    DLMF id
    DLMF:19.7.E9
    0 references
    constraint
    ( 1 - α 2 ) ⁢ ( 1 - ω 2 ) = 1 - k 2 1 superscript 𝛼 2 1 superscript 𝜔 2 1 superscript 𝑘 2 {\displaystyle{\displaystyle(1-\alpha^{2})(1-\omega^{2})=1-k^{2}}}
    0 references
    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    test
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2aadec
    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.m2addec
    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.m2addec
    0 references
     
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