Here's something which is likely mathematically 'trivial'(adsbygoogle = window.adsbygoogle || []).push({});

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.

Thanks in advance for any elucidations about this!

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

Can you offer guidance or do you also need help?

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