Brooks (brooksmoses) wrote,

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Need an equation....

Here's a math puzzle that I'd like a solution to -- having an answer to it would make a fairly key part of my thesis work much easier.

I need an function z=f(x, y) for a surface in three-space with the following characteristics:
  • The surface is, essentially, a bump; it has a finite maximum at (x,y) = (0,0), and decreases monotonically from there down to zero as the radius increases.
  • The surface is defined and continuous everywhere.
  • The first-order derivatives of the surface are defined and continuous everywhere, specifically including the origin.
  • The surface is radially symmetric; it can be expressed exactly as z=g(x2+y2).
  • The surface has compact support; there exists some radius R such that, for x2+y2>R, z=0.

That, as it stands, is the easy part; it suffices to explain the general character of what I'm looking for. I have one final requirement, which makes this a far more difficult problem:
  • The integral of this function f over any given rectangle in xy-space must have a relatively simple closed-form solution. Integrals involving special functions (e.g., Bessel functions) are acceptable if there exists a simple and accurate numerical algorithm for computing them.

Suggestions welcomed. Needless to say, a proof that a solution does not exist will also be useful.

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