Can someone please explain the connection between electrostatic potential and Laplace's equation? And what does that have to do with harmonic functions?
In my opinion you're best advised to consult any decent undergraduate textbook on electro-magnetism and study the salient chapters at your leisure.
eg. Lorraine and Corson - Electromagnetic Fields and Waves, Jackson -Classical Electrodynamics.
In a nutshell:
1. In regions of space where there is no charge density the electrostatic potential satisfies Laplaces equation: where V is the electrostatic potential.
2. Any real function with continuous second partial derivatives which satisfies Laplace's equation is called a harmonic function.
Background knowledge, derivation, details and examples (and answers to the million and one questions the nutshell has prompted) are what constitute the above-mentioned chapters.
Just curious by the way ...... how do you happen to be in the unfortunate circumstance of this question being urgent homework?