DLMF:19.4.E4 (Q6124)

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DLMF:19.4.E4
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    ∂ ⁡ Π ⁡ ( α 2 , k ) ∂ ⁡ k = k k ′ 2 ⁢ ( k 2 - α 2 ) ⁢ ( E ⁡ ( k ) - k ′ 2 ⁢ Π ⁡ ( α 2 , k ) ) . partial-derivative complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝑘 𝑘 superscript superscript 𝑘 ′ 2 superscript 𝑘 2 superscript 𝛼 2 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\frac{\partial\Pi\left(\alpha^{2},k\right)}{% \partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{% \prime}}^{2}\Pi\left(\alpha^{2},k\right)).}}
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    DLMF:19.4.E4
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