DLMF:19.5.E1 (Q6130)

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DLMF:19.5.E1
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    K ⁡ ( k ) = π 2 ⁢ ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k 2 ⁢ m = π 2 ⁢ F 1 2 ⁡ ( 1 2 , 1 2 ; 1 ; k 2 ) , complete-elliptic-integral-first-kind-K 𝑘 𝜋 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript 𝑘 2 𝑚 𝜋 2 Gauss-hypergeometric-F-as-2F1 1 2 1 2 1 superscript 𝑘 2 {\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}% \frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k% ^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)% ,}}
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    DLMF:19.5.E1
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