DLMF:19.30.E6 (Q6680)

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DLMF:19.30.E6
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    Statements

    Defining formula
    ∂ ⁡ s ∂ ⁡ ( 1 / k ) = a 2 - b 2 ⁢ F ⁡ ( ϕ , k ) = a 2 - b 2 ⁢ R F ⁡ ( c - 1 , c - k 2 , c ) , partial-derivative 𝑠 1 𝑘 superscript 𝑎 2 superscript 𝑏 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 superscript 𝑎 2 superscript 𝑏 2 Carlson-integral-RF 𝑐 1 𝑐 superscript 𝑘 2 𝑐 {\displaystyle{\displaystyle\frac{\partial s}{\partial(1/k)}=\sqrt{a^{2}-b^{2}% }F\left(\phi,k\right)=\sqrt{a^{2}-b^{2}}R_{F}\left(c-1,c-k^{2},c\right),}}
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    DLMF id
    DLMF:19.30.E6
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    constraint
    c = csc 2 ⁡ ϕ 𝑐 2 italic-ϕ {\displaystyle{\displaystyle c={\csc^{2}}\phi}}
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    k 2 = ( a 2 - b 2 ) / ( a 2 + λ ) superscript 𝑘 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑎 2 𝜆 {\displaystyle{\displaystyle k^{2}=(a^{2}-b^{2})/(a^{2}+\lambda)}}
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    c = csc 2 ⁢ ϕ 𝑐 2 italic-ϕ {\displaystyle{\displaystyle c={{\csc^{2}}}\phi}}
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    Symbols used
    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.m2aadec
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    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.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.m2adec
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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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