Complete elliptic integral of the first kind

Remarks

The second integral comes from the first by $t = \sin\theta$; it is an improper integral at $t = 1$ that converges. Example: $K(0) = \int_0^{\pi/2} d\theta = \pi/2$. Expanding $(1 - k^2\sin^2\theta)^{-1/2}$ by the binomial series and integrating term by term gives $$K(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left( \frac{(2n)!}{2^{2n} (n!)^2} \right)^2 k^{2n}.$$ Non-example: the formula does not define a finite number at $k = 1$, since $\int_0^{\pi/2} d\theta / \cos\theta$ diverges; indeed $K(k) \to \infty$ as $k \to 1^-$. Some authors use the parameter $m = k^2$ and write $K(m)$; the value is the same function written in another variable. The arithmetic–geometric mean gives $K(k) = \pi / (2 M(1, k'))$.