DLMF:19.4.E6 (Q6126)

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DLMF:19.4.E6
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    Statements

    Defining formula
    ∂ ⁡ E ⁡ ( ϕ , k ) ∂ ⁡ k = E ⁡ ( ϕ , k ) - F ⁡ ( ϕ , k ) k , partial-derivative elliptic-integral-second-kind-E italic-ϕ 𝑘 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=% \frac{E\left(\phi,k\right)-F\left(\phi,k\right)}{k},}}
    0 references
    DLMF id
    DLMF:19.4.E6
    0 references
    Symbols used
    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
    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.m2aadec
    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.m4abdec
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    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2abdec
    0 references
    Q11807
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1adec
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