Brooks (brooksmoses) wrote,
Brooks
brooksmoses

  • Mood:

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.
Subscribe
  • Post a new comment

    Error

    default userpic

    Your reply will be screened

    Your IP address will be recorded 

    When you submit the form an invisible reCAPTCHA check will be performed.
    You must follow the Privacy Policy and Google Terms of use.
  • 27 comments