DLMF:19.25.E4 (Q7204)

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DLMF:19.25.E4
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    Statements

    DLMF defining formula
    Π ⁡ ( α 2 , k ) = - 1 3 ⁢ ( k 2 / α 2 ) ⁢ R J ⁡ ( 0 , 1 - k 2 , 1 , 1 - ( k 2 / α 2 ) ) , complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 1 3 superscript 𝑘 2 superscript 𝛼 2 Carlson-integral-RJ 0 1 superscript 𝑘 2 1 1 superscript 𝑘 2 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=-\tfrac{1}{3}(k^{2}/% \alpha^{2})R_{J}\left(0,1-k^{2},1,1-(k^{2}/\alpha^{2})\right),}}
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    constraint
    - ∞ < k 2 < 1 < α 2 superscript 𝑘 2 1 superscript 𝛼 2 {\displaystyle{\displaystyle-\infty<k^{2}<1<\alpha^{2}}}
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    Symbols used
    symmetric elliptic integral of third kind
    DLMF 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.m2aadec
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    Legendre’s complete elliptic integral of the third kind
    DLMF defining formula
    Π ⁡ ( α 2 , k ) complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m4abdec
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    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1cdec
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    real or complex parameter
    DLMF defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1bdec
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    Identifiers

     
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