DLMF:19.30.E3 (Q6676)

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DLMF:19.30.E3
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    Statements

    Defining formula
    s / a = E ⁡ ( ϕ , k ) = R F ⁡ ( c - 1 , c - k 2 , c ) - 1 3 ⁢ k 2 ⁢ R D ⁡ ( c - 1 , c - k 2 , c ) , 𝑠 𝑎 elliptic-integral-second-kind-E italic-ϕ 𝑘 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 1 3 superscript 𝑘 2 Carlson-integral-RD 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},% c\right)-\tfrac{1}{3}k^{2}R_{D}\left(c-1,c-k^{2},c\right)},}}
    0 references
    DLMF id
    DLMF:19.30.E3
    0 references
    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.m2adec
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    symmetric elliptic integral of first kind
    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.m2adec
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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.m2adec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1bdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1adec
    0 references
    Q11915
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C19.S30.XMD1.m1adec
    0 references
     
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