K(m)=(pi/2)2F1(1/2,1/2;1;m)
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Theorem
The following formula holds: $$K(m)=\dfrac{\pi}{2} {}_2F_1 \left( \dfrac{1}{2}, \dfrac{1}{2}; 1; m \right),$$ where $K$ denotes Elliptic K, $\pi$ denotes pi, and ${}_2F_1$ denotes hypergeometric 2F1.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $17.3.9$
