# Math Help - Two Dimensional Laplace Equation

1. ## Two Dimensional Laplace Equation

Given,
u_xx + u_yy = 0
We know by the Cauchy-Riemann equations that if f(z) is analytic on D then Re{f(x)} and Im{f(z)} are solutions to the Laplace equation on D.

Is the converse true? And furthermore, if the converse is true can we give an explicit formula to solve Laplace equation satisfing the Dirichlet problem?

2. Originally Posted by ThePerfectHacker
We know by the Cauchy-Riemann equations that if f(z) is analytic on D then Re{f(x)} and Im{f(z)} are solutions to the Laplace equation on D. Is the converse true?
Not quite sure of what you are asking. Is this it?

There is rather famous example: u(x,y)=x^2 – y^2 and v(x,y)=2xy.
Both are harmonic, because they are the standard u+iv decomposition of z^2.
Moreover, v is the harmonic conjugate of u. BUT u is not the harmonic conjugate of v, because 2xy+i(x^2 - y^2) is nowhere analytic.

3. Originally Posted by Plato
Not quite sure of what you are asking. Is this it?
No.

If f(z) is analytic then Re{f(z)} and Im{f(z)} are harmonic.

Converse (not exactly),

If u(x,y) is a solution to the 2 Dimensional Laplace Equation (harmonic), then u = Re{f(z)} or Im{f(z)} for some analytic complex function f(z).

4. Originally Posted by ThePerfectHacker
If u(x,y) is a solution to the 2 Dimensional Laplace Equation (harmonic), then u = Re{f(z)} or Im{f(z)} for some analytic complex function f(z).
Isn’t that exactly the counterexample I gave you.
Let u(x,y)=2xy and v(x,y)=x^2 – y^2.
Both are harmonic! BUT f(z)=2xy+i(x^2 - y^2) is nowhere analytic.

5. Originally Posted by Plato
Isn’t that exactly the counterexample I gave you.
Let u(x,y)=2xy and v(x,y)=x^2 – y^2.
Both are harmonic! BUT f(z)=2xy+i(x^2 - y^2) is nowhere analytic.
Yes, thank you.