The Gauss-Conform projection arises from specifying several constraints.
The conform constraint specifies that angles on the surface of the earth must be the same as on the map. This implies a set of coupled real valued PDE's.
By a simple change of variable, the coupled PDE's are shown to be the Cauchy-Riemann equations. Thus ANY analytic function of the new variable set satifies the conformal constraint.
Therefore we were free to choose an analytic function that matched the other constraints. Namely that the X value be zero on the central meridian and the Y value on the central merdian be the length of the central meridian arc from the equator to the point.
By generalizing the real valued function, the length of the central meridian arc, to an analytic complex valued function, we found an analytic function that satisfied the boundary conditions.
Thereafter to perform the mapping all that was needed was a numeric procedure for calculating the value of the function.