# Poisson's and Laplace's equation

1. Dec 2, 2011

### roshan2004

We can easily derive Poisson's and Laplace's equations in electrostatics by using Gauss's law. However, my question is what are the importance of these equations in Electrostatics ?

2. Dec 2, 2011

### abr_pr90

Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful, e.g., solutions to complex problems can be constructed by summing simple solutions.

$\nabla ^{2} V = 0$

So

The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the wave equation, which generally have less regularity.

http://en.wikipedia.org/wiki/Laplace's_equation