Laplace’s Equation

The preceding conditions impose the following conditions on the field equation:

• Since there is no action-at-a-distance, but a field has to have some sort of spacial interaction, the equation can only be a differential equation.

• The equation ought to be of the lowest possible order because, the higher the order of a differential equation, the more its solution is affected by random noise.

• An odd-order differential equation violates the condition of isotropy.

The homogeneous equation that best satisfies these conditions is Laplace’s equation,

Ñ2 φ (x, y) = 0,

(E1)

in which x and y are the Cartesian coordinates of a point in the image.

In two dimensions, as in an image, there is a powerful method to solve Laplace’s equation.  We represent each point, (x, y), by the complex number

z = x + i y,

(E2)

in which i is the square root of −1.  Then, any function of z that is analytic (i.e. satisfies continuity conditions) is a solution of Laplace’s equation.

These requirements, along with some additional reasoning that is also not mathematically rigorous, lead to an equation known as “Laplace’s Equation”.  This equation applies to many physical fields.

In two dimensions, as in an image, Laplace’s equation is easy to solve.  Almost any algebraic expression can be made into a solution.