K(m)=(pi/2)2F1(1/2,1/2;1;m)

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.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $17.3.9$