Weierstraß data

A minimal surface can be described often conveniently in terms of its Weierstraß data.
For Costa's surface, these consist of a meromorphic function G (the Gauß map) and a meromorphic height differential dh on the square torus. Given these data, a surface parametrization is obtained by the formula

The use of elliptic functions can be avoided if we allow for multivalued functions instead:

Here the Gauß map G is multivalued, its natural holomorphic continuation is defined on a square torus. For the moment, we restrict the domain of definition of G and dh to the first quadrant.

Without going into to much detail, we point out that there are four points where the Gauß map is singular, namely at 0,1,-1 and ∞. The point 0 corresponds to the central symmetry point of the surface where the coordinate axes meet. The points 1 and -1 correspond to the catenoidal ends, while ∞ corresponds to the middle planar end. The behaviour of a minimal surface at the ends is completely determined by the singularities of G and dh at these points.

We introduce the 1-forms

These can be integrated in terms of hypergeometric functions:

To avoid problems with branch cuts later, we will use instead:

To do the integration with Mathematica, it is advisable to rewrite first

Then indeed