DLMF:23.5#Ex12 (Q7240)

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DLMF:23.5#Ex12
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    Defining formula
    K ⁡ ( k ) = e i ⁢ π / 6 ⁢ K ′ ⁡ ( k ) = e i ⁢ π / 12 ⁢ 3 1 / 4 ⁢ ( Γ ⁡ ( 1 3 ) ) 3 2 7 / 3 ⁢ π . complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑒 imaginary-unit 𝜋 6 diffop complete-elliptic-integral-first-kind-K 1 𝑘 superscript 𝑒 imaginary-unit 𝜋 12 superscript 3 1 4 superscript Euler-Gamma 1 3 3 superscript 2 7 3 𝜋 {\displaystyle{\displaystyle K\left(k\right)=e^{\mathrm{i}\pi/6}K'\left(k% \right)=e^{\mathrm{i}\pi/12}\frac{3^{1/4}\left(\Gamma\left(\frac{1}{3}\right)% \right)^{3}}{2^{7/3}\pi}.}}
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    DLMF:23.5#Ex12
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