A formula for the potential of a polygon with varying charge density

Explicit formulae are derived giving the Newtonian potential and field of a polygon whose surface charge density is a linear or quadratic function. The formulae involve only the elementary functions, logarithm and arctangent. Special attention is given to triangles and rectangles because they appear in various finite element calculations. The formulae may be used, together with the Fredholm integral equation, to solve electric or magneticfield problems.