DLMF:19.25.E11 (Q6515)

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DLMF:19.25.E11
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    Statements

    Defining formula
    E ⁡ ( ϕ , k ) = - 1 3 ⁢ k ′ 2 ⁢ R D ⁡ ( c - k 2 , c , c - 1 ) + ( c - k 2 ) / ( c ⁢ ( c - 1 ) ) , elliptic-integral-second-kind-E italic-ϕ 𝑘 1 3 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 𝑐 superscript 𝑘 2 𝑐 𝑐 1 𝑐 superscript 𝑘 2 𝑐 𝑐 1 {\displaystyle{\displaystyle E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2% }R_{D}\left(c-k^{2},c,c-1\right)+\sqrt{(c-k^{2})/(c(c-1))},}}
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    DLMF id
    DLMF:19.25.E11
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    constraint
    ϕ ≠ 1 2 ⁢ π italic-ϕ 1 2 𝜋 {\displaystyle{\displaystyle\phi\neq\tfrac{1}{2}\pi}}
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    Symbols used
    elliptic integral symmetric in only two variables
    Defining formula
    R D ⁡ ( x , y , z ) Carlson-integral-RD 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E5.m2addec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    Legendre’s incomplete elliptic integral of the second kind
    Defining formula
    E ⁡ ( ϕ , k ) elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E5.m2addec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1fdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1jdec
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    Q11815
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1ddec
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