DLMF:19.25.E16 (Q6520)

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DLMF:19.25.E16
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    Statements

    Defining formula
    Π ⁡ ( ϕ , α 2 , k ) = - 1 3 ⁢ ω 2 ⁢ R J ⁡ ( c - 1 , c - k 2 , c , c - ω 2 ) + ( c - 1 ) ⁢ ( c - k 2 ) ( α 2 - 1 ) ⁢ ( 1 - ω 2 ) ⁢ R C ⁡ ( c ⁢ ( α 2 - 1 ) ⁢ ( 1 - ω 2 ) , ( α 2 - c ) ⁢ ( c - ω 2 ) ) , elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 1 3 superscript 𝜔 2 Carlson-integral-RJ 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 superscript 𝜔 2 𝑐 1 𝑐 superscript 𝑘 2 superscript 𝛼 2 1 1 superscript 𝜔 2 Carlson-integral-RC 𝑐 superscript 𝛼 2 1 1 superscript 𝜔 2 superscript 𝛼 2 𝑐 𝑐 superscript 𝜔 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)=-\tfrac{1}{3}% \omega^{2}R_{J}\left(c-1,c-k^{2},c,c-\omega^{2}\right)+\sqrt{\frac{(c-1)(c-k^{% 2})}{(\alpha^{2}-1)(1-\omega^{2})}}\*R_{C}\left(c(\alpha^{2}-1)(1-\omega^{2}),% (\alpha^{2}-c)(c-\omega^{2})\right),}}
    0 references
    DLMF id
    DLMF:19.25.E16
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    constraint
    ω 2 = k 2 / α 2 superscript 𝜔 2 superscript 𝑘 2 superscript 𝛼 2 {\displaystyle{\displaystyle\omega^{2}=k^{2}/\alpha^{2}}}
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    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    Defining formula
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2adec
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    symmetric elliptic integral of third kind
    Defining formula
    R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}
    xml-id
    C19.S16.E2.m2acdec
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    Legendre’s incomplete elliptic integral of the third kind
    Defining formula
    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.m2abdec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1kdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1odec
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    Q11809
    Defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1edec
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