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

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- Dec 18th 2006, 09:09 AMtaypezComplex 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 - Dec 18th 2006, 10:26 AMThePerfectHacker

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. - Dec 18th 2006, 11:15 AMPlato
**The function is analyic, hence both u and v are**__harmonic__.

It satisfies the Cauchy-Riemann Equations.

&

Now,

Add them,

because

&

But - Dec 18th 2006, 01:33 PMThePerfectHacker
- Dec 18th 2006, 02:16 PMPlato
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 .