DLMF:19.7#Ex7 (Q6188)

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DLMF:19.7#Ex7
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    E ⁡ ( 1 / k ) = ( 1 / k ) ⁢ ( E ⁡ ( k ) ± i ⁢ E ⁡ ( k ′ ) - k ′ 2 ⁢ K ⁡ ( k ) ∓ i ⁢ k 2 ⁢ K ⁡ ( k ′ ) ) , complete-elliptic-integral-second-kind-E 1 𝑘 1 𝑘 minus-or-plus plus-or-minus complete-elliptic-integral-second-kind-E 𝑘 imaginary-unit complete-elliptic-integral-second-kind-E superscript 𝑘 ′ superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 imaginary-unit superscript 𝑘 2 complete-elliptic-integral-first-kind-K superscript 𝑘 ′ {\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)\pm% \mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)\mp\mathrm{i% }k^{2}K\left(k^{\prime}\right)\right),}}
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    DLMF:19.7#Ex7
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