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

  1. Jul 12, 2007 #1

    xez

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    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.

    Thanks in advance for any elucidations about this!
     
  2. jcsd
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