Thread: Complex Analysis, harmonic functions

Here is the problem:

If f= u+ iv is analytic in a region, show that uv is harmonic in the region but that u^2 need not be harmonic.

Thanks

Originally Posted by taypez

Here is the problem:

If f= u+ iv is analytic in a region, show that uv is harmonic in the region but that u^2 need not be harmonic.

Thanks

The function is analyic, hence homolorphic.
It satisfies the Cauchy-Riemann Equations.


Thus,


We need to show,

Solves the PDE,
.

Now,


Add them,

But from Cauchy-Riemann Equations,

Maybe there is some theorem that tell the right hand side is zero. I do not know of one.

The function is analyic, hence both u and v are harmonic.
It satisfies the Cauchy-Riemann Equations.
&
Now,


Add them,
because
&

But

Originally Posted by Plato

The function is analyic, hence both u and v are harmonic.

Slow down, I did not get that. You are saying that if a complex function is analytic (meaning it has a power series expansion, right?) then it must be harmonic, (satisfy the Laplace equation). So that was the extra equation you used to complete the proof?

Originally Posted by ThePerfectHacker

a complex function is analytic (meaning it has a power series expansion, right?) then it must be harmonic, (satisfy the Laplace equation).

For the complex analyst the definition is: a function said to be analytic at if its derivative exists at each point in some neighborhood of .

Moreover, a function is said to be harmonic if , that is the Laplace equation as you have noted.

Theorem: If is analytic is the domain , then both and are harmonic in .