REAL Removable singularities, 0/0, numeric evaluation, limits, partial deriv.'s

1. Jul 12, 2007

xez

Here's something which is likely mathematically 'trivial'
in an analytic sense, but which is leading to some
difficulties for me as part of a much larger discrete
problem so I thought I'd see if someone could
help in this aspect.

The problem is to evaluate the usefully correct value of
these equations (coordinate mappings) and each of
their first partial derivatives at the singular values.

The equations are of real, scalar valued continuous
variables over the following domains:

-1 <= x,e,z,r,s <= +1
though x,e,z are NOT linearly independent and they
range over a smaller space than an [-1:+1] axis cube.

The mapping equations are:
r=( -2*(1+x)/(e+z))-1
s=(+2*(1+e)/(1-z))-1

There are singularities at every locus where (e+z) == 0,
and only as a subset of those when z==+1 and e=-1.

In the discrete cases of interest in the x;e;z sub domain,
the singularities occur only where:
e == -z, and simultaneously x == -1.

In all cases of interest for the above mapping equations,
I believe that the problem is a real-valued interpretation
of a limit as the numerator and denominator both
approach 0, so the result is 0/0 at the singularity.

The equations yield well defined differentiable
values over the deleted (real) neighborhood of
each singularity. So were these complex domain
equations, the singularities should be removable.

The partials are as follows, and run into similar problems:
dr/dx=-2/(z+e)
dr/de=2*(x+1)/(z+e)^2
dr/dz=2*(x+1)/(z+e)^2
ds/de=2/(1-z)
ds/dz=2*(e+1)/(1-z)^2

So, the question/problem is:

Where e=-z, and x=-1, define:
r(x,e,z)
s(e,z)
dr/dx,
dr/de
dr/dz
ds/de
ds/dz

Removing the singularity is well known wrt. integration,
and with respect to complex functions,
though the theory and practice of doing it with respect
to discrete evaluation and partial differentiation of
real valued functions of multiple non-orthogonal
variables isn't as familiar to me.

In this case, it seems like the values ought
to be easily determined analytically as limits.

Most of all, I'm trying to get it simplified so that
when discretely calculated the result is found without
using numerically unstable evaluations of the
function with small coordinate deltas to evaluate the
value of the function in the deleted neighborhood of
the singularity.

I believe I may be looking for either a constant limit
or some kind of change-of-variable type of evaluation
that lets me evaluate the value without numerically
undesirable calculations involving small deltas or
ratios of small numbers.

I suppose it's possible to define a finite series to well
interpolate the value of the function throughout the
neighborhood of the singularity even though
analytic continuation in general doesn't apply
(AFAIK) in the real domain.