DLMF:19.30.E6 (Q6680)

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DLMF:19.30.E6
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    Statements

    test
    ∂ ⁡ 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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