- #1
Fosheimdet
- 15
- 2
If $$f(z)=u(x,y)+iv(x,y)$$ is analytic in a domain D, then both u and v satisfy Laplace's equations
$$\nabla^2 u=u_{xx} + u_{yy}=0$$
$$\nabla^2 v=v_{xx} + v_{yy}=0$$
and u and v are called harmonic functions.
My question is whether or not this goes both ways. If you have two functions u and v which satisfy the Laplace equations are they the real and imaginary parts of an analytic function?
$$\nabla^2 u=u_{xx} + u_{yy}=0$$
$$\nabla^2 v=v_{xx} + v_{yy}=0$$
and u and v are called harmonic functions.
My question is whether or not this goes both ways. If you have two functions u and v which satisfy the Laplace equations are they the real and imaginary parts of an analytic function?