# Determine whether functions are harmonic

1. Apr 6, 2015

### Shackleford

1. The problem statement, all variables and given/known data

Determine whether or not the following functions are harmonic:

$u = z + \bar{z}$

$u = 2z\bar{z}$

2. Relevant equations

$z = u(x,y) + v(x,y)i$

$\bar{z} = u(x,y) - v(x,y)i$

A function is harmonic if Δu = 0.

3. The attempt at a solution

$Δu = Δz +Δ \bar{z} = u_{xx} + v_{xx} + u_{yy} + v_{yy} + u_{xx} - v_{xx} + u_{yy} + -v_{yy} = 2u_{xx} + 2u_{yy}$

$u = 2z\bar{z} = 2[u(x,y) + v(x,y)i][u(x,y) - v(x,y)i] = 2[u^2(x,y) - v^2(x,y)]$

$Δu = 2[2uu_{xx} + 2uu_{yy} - 2vv_{xx} - 2vv_{yy}]$

Last edited: Apr 6, 2015
2. Apr 6, 2015

### Dick

I don't see why you are struggling with this. $z=x+iy$. If $u=z+\bar{z}$ then $u(x,y)=2x$. Is that harmonic?

3. Apr 6, 2015

### Shackleford

I wanted to use the more general case. To be honest, I just wanted to check my work.

If it's not zero, then it's not harmonic.

4. Apr 6, 2015

### Dick

I'm not sure what you are saying here. Don't do the general case. Just do these two special cases. What about those?

5. Apr 6, 2015

### Shackleford

Sorry. It was my mistake. For some reason I wanted to generalize to a function f(z).

Here, the first is harmonic. Δ(2x) = 0 and Δ(2x2+2y2) = 4 + 4 = 8.

6. Apr 6, 2015

### Dick

$(x+iy)(x-iy)$ is not equal to $x^2-y^2$.

7. Apr 6, 2015

Corrected.

8. Apr 6, 2015

### Dick

That's better.

9. Apr 6, 2015

### Shackleford

Thanks again.