Transcendental Values of Some Elliptic Integrals

Abstract

In the case of trigonometric functions, we can rewrite the familiar identity

$$\displaystyle{\sin ^{2}z +\cos ^{2}z = 1}$$

as

$$\displaystyle{y^{2} + \left ({dy \over dz}\right )^{2} = 1}$$

where y(z) = sinz. We can retrieve the inverse function of sine by formally integrating

$$\displaystyle{dz ={ dy \over \sqrt{1 - y^{2}}},}$$

so that

$$\displaystyle{\sin ^{-1}z =\int _{ 0}^{z}{ dy \over \sqrt{1 - y^{2}}}.}$$

The period of the sine function can also be retrieved from

$$\displaystyle{2\pi = 4\int _{0}^{1}{ dy \over \sqrt{1 - y^{2}}}.}$$

However, we should be cautious about this reasoning since sin⁻¹ z is a multi-valued function and the integral may depend on the path taken from 0 to z. With this understanding, let us try to treat the inverse of the elliptic function $\wp (z)$ in a similar way. Indeed, we have

$$\displaystyle{{d\wp (z) \over dz} = \sqrt{4\wp (z)^{3 } - g_{2 } \wp (z) - g_{3}}}$$