Elliptic integral

Remarks

The substitution $t = \sin\theta$ turns $F(\varphi, k)$ into $\int_0^{\sin\varphi} dt / \sqrt{(1 - t^2)(1 - k^2 t^2)}$, an elliptic integral with $P(t) = (1 - t^2)(1 - k^2 t^2)$, which has the four distinct roots $\pm 1$ and $\pm 1/k$ when $0 < k < 1$. Example: the ellipse $x = a\sin\theta$, $y = b\cos\theta$ with $0 < b < a$ has arc length element $\sqrt{a^2\cos^2\theta + b^2\sin^2\theta}\,d\theta = a\sqrt{1 - e^2\sin^2\theta}\,d\theta$, where $e^2 = 1 - b^2/a^2$; so its arc length from $\theta = 0$ to $\theta = \varphi$ is $a E(\varphi, e)$. Non-example: $\int_0^x dt / \sqrt{1 - t^2} = \arcsin x$ is not an elliptic integral, since $P(t) = 1 - t^2$ has degree $2$. Every elliptic integral is a sum of elementary functions and constant multiples of integrals of the three Legendre kinds. The values at $\varphi = \pi/2$ are the complete integrals $K(k) = F(\pi/2, k)$, of the first kind, and $E(k) = E(\pi/2, k)$, of the second kind.