Asymptotic coordinates: Because this series cannot represent the whole surface, I tried a coordinate system suggested by the proof of Hilbert's non-existence theorem. At each point of a negative surface in three-space, since the curvatures go from positive to negative, there are two directions of zero curvature. Following these two directions gives a grid that defines a coordinate system. This is the coordinate system in the illustration referenced above. Here are the series versions of the coordinates.
Some properties: The coordinates for the grid can be chosen so that every coordinate curve is parametrized by arclength, and the angle θ between the two coordinate directions satisfies the differential equation θxy = sin(θ). This coordinate system preserves two facets of the original surface z = xy; we can keep the same origin and axes, and θ depends only on xy.
Intro Curvature −1 Coordinates θ Complex θ More Equationss Trick General Entire