Elliptic Integrals and Hypergeometric Series

We have already discussed briefly the problem of inversion for the Weierstrass ℘-function. In this way, we were able to recover the transcendental nature of the periods whenever the invariants g 2 , g 3 $$g_{2},g_{3}$$ were algebraic. We now look at the calculation a bit more closely. Before we begin, it may be instructive to look at a familiar example. Clearly, we have b = ∫ 0 sin b d y 1 − y 2 . $$\displaystyle{b =\int _{ 0}^{\sin b}{ dy \over \sqrt{1 - y^{2}}}.}$$ But how should we view this equation? Since sinb is periodic with period 2π, we can only view this as an equation modulo 2π. If sinb is algebraic, then, we know as a consequence of the Hermite–Lindemann Hermite–Lindemann theorem theorem that b is transcendental. In this way, we deduce that the integral ∫ 0 α d y 1 − y 2 $$\displaystyle{\int _{0}^{\alpha }{ dy \over \sqrt{1 - y^{2}}}}$$ is transcendental whenever α is a non-zero algebraic number in the interval [−1, 1].