By transforming the problem into one which the rich theory of complex variables can be applied, some important gains can be made. It is also possible to proceed by solving the coupled real valued PDE's (48) and (49) by a tedious and tricky ``ladder'' approach. The result is a power series in latitude and longitude that unfortunately converges too slowly for practical use at high latitudes.
The route from here on is to transform by change of variable the PDE's into the Cauchy-Riemann equations. This shows that the mapping function is in fact, (in terms of the new variable), an analytic complex valued function. All that remains therefore is to find any analytic function that satisfies the boundary constraints.
If a complex valued function satisfies the Cauchy-Riemann equations in a region and the partial derivatives are continous in that region, then R is analytic in that region. [Spiegel p63]
The Cauchy-Riemann equations are...
Notice first that the basic PDE's (48) and (49) of the Gauss-Conform projection look very similar to the Cauchy-Riemann equations, but are multiplied by an additional term.
However one can transform, by change of variables, the basic PDE's so that they DO
satisfy the Cauchy-Riemann equations. Define such that...
Then the basic PDE's in terms of become exactly the Cauchy-Riemann equations. Thus R is an analytic function of .
All we need to do is choose a function R, which is analytic function in the region we are interested, that satisfies the boundary condition.
Now our boundary condition constraints are and . Where L is the distance along the central meridian from the equator. Which translates to .
First define the complex generalization of ...
Where l is complex and the integral sign denotes complex integration. Then from a theorem on page 97 of [Spiegel] we know that is analytic in the same region as the integrand.
We also know that the inverse function, which I will denote by t( l) and define implicitly by , is analytic in the same region as except where the derivative of is zero.
Now define the complex generalisation of L from (55) as
As above, the integral is analytic. We know that t(l) is analytic, thus from the chain rule for complex differentiation we know that L(l) is analytic as it is a composite of two analytic functions.
We can check that this satisfies the boundary conditions...
And here, in the choice of R, is I suspect is the real reason for slightly strange South African LO coordinate system. The reason the LO X coordinate pointed South was to spare the early surveyor the effort and philosophical pain of having to multiply by the square root of minus one in two places.