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.