Exact, approximate and asymptotic solutions of the Klein-Gordon integral equation

Interaction of the highly localized probe of scanning probe microscopy with solid surfaces with mobile electronic or ionic carriers leads to the redistribution of mobile carriers at the tip surface junction. For small probe biases, this problem is equivalent to the Debye screening, described by Klein-Gordon (K-G) integral equation. Here, an exact solution to the K-G equation is derived for the case of a circle in the form of a convergent series expansion of the solution, which is effective for relatively small values of the inverse Debye length, k. Also, a reasonably accurate solution is derived for large values of parameter k by using the method of collocation. A surprisingly simple asymptotic solution is derived for very large values of k, which is valid for the arbitrary right-hand side of the equation. The same methods can be used for the case of elliptic domain.