DLMF:19.25.E2 (Q6506)

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DLMF:19.25.E2
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    Statements

    Defining formula
    Π ⁡ ( α 2 , k ) - K ⁡ ( k ) = 1 3 ⁢ α 2 ⁢ R J ⁡ ( 0 , k ′ 2 , 1 , 1 - α 2 ) . complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 complete-elliptic-integral-first-kind-K 𝑘 1 3 superscript 𝛼 2 Carlson-integral-RJ 0 superscript superscript 𝑘 ′ 2 1 1 superscript 𝛼 2 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)-K\left(k\right)=% \tfrac{1}{3}\alpha^{2}R_{J}\left(0,{k^{\prime}}^{2},1,1-\alpha^{2}\right).}}
    0 references
    DLMF id
    DLMF:19.25.E2
    0 references
    Symbols used
    symmetric elliptic integral of third kind
    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.m2adec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aadec
    0 references
    Q11816
    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.m4adec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1adec
    0 references
    Q11815
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1adec
    0 references
    Q11809
    Defining formula
    α 2 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha^{2}}}
    xml-id
    C19.S1.XMD7.m1dec
    0 references
     
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