DLMF:19.25.E14 (Q6518)

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DLMF:19.25.E14
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    Statements

    test
    Π ⁡ ( ϕ , α 2 , k ) - F ⁡ ( ϕ , k ) = 1 3 ⁢ α 2 ⁢ R J ⁡ ( c - 1 , c - k 2 , c , c - α 2 ) , elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 1 3 superscript 𝛼 2 Carlson-integral-RJ 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)-F\left(\phi,k% \right)=\tfrac{1}{3}\alpha^{2}R_{J}\left(c-1,c-k^{2},c,c-\alpha^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.25.E14
    0 references
    Symbols used
    symmetric elliptic integral of third kind
    test
    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.m2abdec
    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.m2abdec
    0 references
     
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