DLMF:19.25.E25 (Q6531)

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DLMF:19.25.E25
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    Statements

    Defining formula
    ( z - x ) 3 / 2 ⁢ R D ⁡ ( x , y , z ) = ( 3 / k 2 ) ⁢ ( F ⁡ ( ϕ , k ) - E ⁡ ( ϕ , k ) ) , superscript 𝑧 𝑥 3 2 Carlson-integral-RD 𝑥 𝑦 𝑧 3 superscript 𝑘 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 {\displaystyle{\displaystyle(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F% \left(\phi,k\right)-E\left(\phi,k\right)),}}
    0 references
    DLMF id
    DLMF:19.25.E25
    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.m2ahdec
    0 references
    Legendre’s incomplete elliptic integral of the first kind
    Defining formula
    F ⁡ ( ϕ , k ) elliptic-integral-first-kind-F italic-ϕ 𝑘 {\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}
    xml-id
    C19.S2.E4.m2aedec
    0 references
    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.m2afdec
    0 references
    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1odec
    0 references
    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1xdec
    0 references
     
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