DLMF:19.25.E12 (Q6516)

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DLMF:19.25.E12
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    Statements

    Defining formula
    ∂ ⁡ E ⁡ ( ϕ , k ) ∂ ⁡ k = - 1 3 ⁢ k ⁢ R D ⁡ ( c - 1 , c - k 2 , c ) . partial-derivative elliptic-integral-second-kind-E italic-ϕ 𝑘 𝑘 1 3 𝑘 Carlson-integral-RD 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=-% \tfrac{1}{3}kR_{D}\left(c-1,c-k^{2},c\right).}}
    0 references
    DLMF id
    DLMF:19.25.E12
    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.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.m2aedec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4aadec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2aadec
    0 references
    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1gdec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1kdec
    0 references
     
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