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*(*x*^{2}+*y*^{2}). - The surface has compact support; there exists some radius
*R*such that, for*x*^{2}+*y*^{2}>*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.