DLMF:19.25.E6 (Q6510)

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DLMF:19.25.E6
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    Statements

    Defining formula
    ∂ ⁡ F ⁡ ( ϕ , k ) ∂ ⁡ k = 1 3 ⁢ k ⁢ R D ⁡ ( c - 1 , c , c - k 2 ) . partial-derivative elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 1 3 𝑘 Carlson-integral-RD 𝑐 1 𝑐 𝑐 superscript 𝑘 2 {\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}=% \tfrac{1}{3}kR_{D}\left(c-1,c,c-k^{2}\right).}}
    0 references
    DLMF id
    DLMF:19.25.E6
    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.m2aadec
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    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.m2aadec
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    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.m4adec
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    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2adec
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    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1adec
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    Q11808
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C19.S1.XMD5.m1edec
    0 references
     
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