DLMF:19.25.E10 (Q7210)

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DLMF:19.25.E10
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    Statements

    DLMF defining formula
    E ⁡ ( ϕ , k ) = k ′ 2 ⁢ R F ⁡ ( c - 1 , c - k 2 , c ) + 1 3 ⁢ k 2 ⁢ k ′ 2 ⁢ R D ⁡ ( c - 1 , c , c - k 2 ) + k 2 ⁢ ( c - 1 ) / ( c ⁢ ( c - k 2 ) ) , elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript superscript 𝑘 ′ 2 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 1 3 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 Carlson-integral-RD 𝑐 1 𝑐 𝑐 superscript 𝑘 2 superscript 𝑘 2 𝑐 1 𝑐 𝑐 superscript 𝑘 2 {\displaystyle{\displaystyle E\left(\phi,k\right)={k^{\prime}}^{2}R_{F}\left(c% -1,c-k^{2},c\right)+\tfrac{1}{3}k^{2}{k^{\prime}}^{2}R_{D}\left(c-1,c,c-k^{2}% \right)+k^{2}\sqrt{(c-1)/(c(c-k^{2}))},}}
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    constraint
    c > k 2 𝑐 superscript 𝑘 2 {\displaystyle{\displaystyle c>k^{2}}}
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    Symbols used
    elliptic integral symmetric in only two variables
    DLMF 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.m2acdec
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    symmetric elliptic integral of first kind
    DLMF defining formula
    R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E1.m2addec
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    Legendre’s incomplete elliptic integral of the second kind
    DLMF 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.m2acdec
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    real or complex argument
    DLMF defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1edec
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    real or complex modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1idec
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    complementary modulus
    DLMF defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C19.S1.XMD6.m1cdec
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    Identifiers

     
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