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
$\displaystyle f(x+iy)=u+iv$

The function is analyic, hence homolorphic.

It satisfies the Cauchy-Riemann Equations.

$\displaystyle u_x=v_y$

$\displaystyle u_y=-v_x$

Thus,

$\displaystyle u_xv_x+u_yv_y=0$

We need to show,

$\displaystyle uv$

Solves the PDE,

$\displaystyle f_{xx}+f_{yy}=0$.

Now,

$\displaystyle (uv)_{xx}=(u_xv+uv_x)_x=u_{xx}v+2u_xv_x+uv_{xx}$

$\displaystyle (uv)_{yy}=u_{yy}v+2u_yv_y+uv_{yy}$

Add them,

$\displaystyle (uv)_{xx}+(uv)_{yy}=u_{xx}v+2(u_xv_x+u_yv_y)+uv_{y y}$

But from Cauchy-Riemann Equations,

$\displaystyle (uv)_{xx}+(uv)_{yy}=u_{xx}v+uv_{yy}$

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.

$\displaystyle u_x=v_y$ & $\displaystyle u_y=-v_x$

Now,

$\displaystyle (uv)_{xx}=(u_xv+uv_x)_x=u_{xx}v+2u_xv_x+uv_{xx}$

$\displaystyle (uv)_{yy}=u_{yy}v+2u_yv_y+uv_{yy}$

Add them,

$\displaystyle (uv)_{xx}+(uv)_{yy}=2(u_xv_x+u_yv_y) $ because

$\displaystyle u_{xx}+ u_{yy}=0$ & $\displaystyle v_{xx}+ v_{yy}=0.$

But $\displaystyle u_x v_x + u_y v_y = u_x \left( { - u_y } \right) + u_y \left( { u_x } \right) = 0$ - Dec 18th 2006, 01:33 PMThePerfectHacker
- Dec 18th 2006, 02:16 PMPlato
For the complex analyst the definition is: a function $\displaystyle f$ said to be

*analytic*at $\displaystyle z_0$ if its derivative exists at each point in some neighborhood of $\displaystyle z_0$.

Moreover, a function $\displaystyle u$ is said to be*harmonic*if $\displaystyle u_{xx} + u_{yy} = 0$, that is the Laplace equation as you have noted.

__Theorem__: If $\displaystyle f(z) = u(x,y) + iv(x,y)$ is analytic is the domain $\displaystyle D$, then both $\displaystyle u(x,y)$ and $\displaystyle v(x,y)$ are harmonic in $\displaystyle D$.