DLMF:19.8.E20 (Q6238)

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DLMF:19.8.E20
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    Statements

    Defining formula
    ρ ⁢ Π ⁡ ( ϕ , α 2 , k ) = 4 1 + k ′ ⁢ Π ⁡ ( ψ 1 , α 1 2 , k 1 ) + ( ρ - 1 ) ⁢ F ⁡ ( ϕ , k ) - R C ⁡ ( c - 1 , c - α 2 ) , 𝜌 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 4 1 superscript 𝑘 ′ elliptic-integral-third-kind-Pi subscript 𝜓 1 superscript subscript 𝛼 1 2 subscript 𝑘 1 𝜌 1 elliptic-integral-first-kind-F italic-ϕ 𝑘 Carlson-integral-RC 𝑐 1 𝑐 superscript 𝛼 2 {\displaystyle{\displaystyle\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k% ^{\prime}}\Pi\left(\psi_{1},\alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k% \right)-R_{C}\left(c-1,c-\alpha^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.8.E20
    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.m2aadec
    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.m2addec
    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.m2aadec
    0 references
    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1gdec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1ndec
    0 references
    Q11815
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1kdec
    0 references
    Q11809
    Defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1edec
    0 references
    Q11833
    Defining formula
    k 1 subscript 𝑘 1 {\displaystyle{\displaystyle k_{1}}}
    xml-id
    C19.S8.XMD13.m1bdec
    0 references
    Q11834
    Defining formula
    ψ 1 subscript 𝜓 1 {\displaystyle{\displaystyle\psi_{1}}}
    xml-id
    C19.S8.XMD14.m1bdec
    0 references
    Q11835
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C19.S8.XMD16.m1dec
    0 references
    Q11836
    Defining formula
    α 1 2 superscript subscript 𝛼 1 2 {\displaystyle{\displaystyle\alpha_{1}^{2}}}
    xml-id
    C19.S8.XMD17.m1dec
    0 references
    Q11837
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C19.S8.XMD18.m1dec
    0 references
     
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