Harmonic Functions: Laplace's Equations & Analytic Functions

[Fosheimdet]

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?

 

[ShayanJ]

Just take the real part of one function and the imaginary part of another function. They satisfy Laplace's equation but aren't the real and imaginary parts of a function because they don't satisfy the Cauchy-Riemann conditions.

 

[mathwonk]

A Shyan said, the answer to your question is no, since the real part of a holomorphic function determines the imaginary part. But every harmonic function is locally the real part of a holomorphic function, although not necessarily globally, due to branching behavior that may occur only in the imaginary part. E.g. log(|z|), defined everywhere but z=0, is the real part of log(z), but the imaginary part of log(z), a multiple of arg(z), is only defined locally near non zero values of z.

 

[WWGD]

In yet another way, the Real part of a holomorphic function can only have ( up to a difference by a constant) one
Complex counterpart, so the odds are that two harmonic functions are respectively the Real and Imaginary part of a holomorphic function.

 

[blue_raver22]

Yes, this statement is known as the Cauchy-Riemann equations. If u and v satisfy Laplace's equations, then they must also satisfy the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. Therefore, u and v can be seen as the real and imaginary parts of an analytic function f(z) = u(x,y) + iv(x,y). However, it is important to note that not all functions that satisfy Laplace's equations are analytic. There are some additional conditions that must also be met for a function to be analytic.