DLMF:19.7.E8 (Q6203)

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DLMF:19.7.E8
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    Statements

    Defining formula
    Π ⁡ ( ϕ , α 2 , k ) + Π ⁡ ( ϕ , ω 2 , k ) = F ⁡ ( ϕ , k ) + c ⁢ R C ⁡ ( ( c - 1 ) ⁢ ( 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 elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑐 Carlson-integral-RC 𝑐 1 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)=F\left(\phi,k\right)+\sqrt{c}R_{C}\left((c-1)(c-k^{2}),(c-% \alpha^{2})(c-\omega^{2})\right),}}
    0 references
    DLMF id
    DLMF:19.7.E8
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    constraint
    α 2 ⁢ ω 2 = k 2 superscript 𝛼 2 superscript 𝜔 2 superscript 𝑘 2 {\displaystyle{\displaystyle\alpha^{2}\omega^{2}=k^{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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    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    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.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.m2acdec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1ddec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1gdec
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    Q11809
    Defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1ddec
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    Q11823
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C19.S7.XMD9.m1dec
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