What is a geometrical interpretation of a harmonic function?

Question

More specifically, what geometrical properties does a function need to have, in order to satisfy the laplace equations? (other than it being 2nd order differentiable)

Answer 1

A "geometric" visualization is at

https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/laplacian/v/harmonic-functions

 

 

Answer 2

It must satisfy Laplace's equation in the x-y plane:

 

f = f(x,y)

 

∂²f/∂x² + ∂²f/∂y² = 0

 

In the theory of complex variables, there is a class of functions of complex variables called analytic functions.  To be analytic means the derivatives of the function are well defined.  Analytic functions satisfy a pair of equations called the Cauchy-Riemann conditions.  Consider a complex function F of a complex variable z.

 

z = x + iy

 

F = F(z) = u(x,y) +iv(x,y),

 

where u and v are the real and imaginary parts of F, respectively.  If F(z) is analytic, then  it must satisfy the Cauchy-Riemann conditions:

 

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

 

If these 2 conditions are satisfied, then both u(x,y) and v(x,y) also satisfy Laplace's equation in the x-y plane, and hence both u and y are harmonic functions.  So you can think of a harmonic function as being either the real or imaginary part of a complex function of a complex variable, where the function is analytic.