DLMF:19.8.E14 (Q6225)

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DLMF:19.8.E14
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    Statements

    Defining formula
    2 ⁢ ( k 2 - α 2 ) ⁢ Π ⁡ ( ϕ , α 2 , k ) = ω 2 - α 2 1 + k ′ ⁢ Π ⁡ ( ϕ 1 , α 1 2 , k 1 ) + k 2 ⁢ F ⁡ ( ϕ , k ) - ( 1 + k ′ ) ⁢ α 1 2 ⁢ R C ⁡ ( c 1 , c 1 - α 1 2 ) , 2 superscript 𝑘 2 superscript 𝛼 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript 𝜔 2 superscript 𝛼 2 1 superscript 𝑘 ′ elliptic-integral-third-kind-Pi subscript italic-ϕ 1 superscript subscript 𝛼 1 2 subscript 𝑘 1 superscript 𝑘 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 1 superscript 𝑘 ′ superscript subscript 𝛼 1 2 Carlson-integral-RC subscript 𝑐 1 subscript 𝑐 1 superscript subscript 𝛼 1 2 {\displaystyle{\displaystyle 2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k% \right)=\frac{\omega^{2}-\alpha^{2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}% ^{2},k_{1}\right)+k^{2}F\left(\phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}% \left(c_{1},c_{1}-\alpha_{1}^{2}\right)},}}
    0 references
    DLMF id
    DLMF:19.8.E14
    0 references
    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
    0 references
    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.m2aadec
    0 references
    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.m2adec
    0 references
    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1bdec
    0 references
     
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