Thread: Is u harmonic - differentiate u

1. Is u harmonic - differentiate u

$\displaystyle u = z^3 \bar{z}^5 + \bar{z}^3 z^5$

I'm not sure of how to differentiate z z-conjugate. I used $\displaystyle z = x + iy$ and $\displaystyle \bar{z} = x - iy$ and started to differentiate to get $\displaystyle u_x$. When I get to $\displaystyle u_{xx}$ it's very bloated. Is there a neater way of doing this than my way?

To check whether u is harmonic or not I need to check if $\displaystyle \Delta u = u_{xx} + u_{yy} = 0$

2. Functions involving conjugates are generally not analytic. That means you may just have to do this by brute force (as you've been attempting). I don't see any clever tricks off the top of my head. I'm curious if anyone else comes up with anything quick.

3. Originally Posted by liquidFuzz
$\displaystyle u = z^3 \bar{z}^5 + \bar{z}^3 z^5$

I'm not sure of how to differentiate z z-conjugate. I used $\displaystyle z = x + iy$ and $\displaystyle \bar{z} = x - iy$ and started to differentiate to get $\displaystyle u_x$. When I get to $\displaystyle u_{xx}$ it's very bloated. Is there a neater way of doing this than my way?

To check whether u is harmonic or not I need to check if $\displaystyle \Delta u = u_{xx} + u_{yy} = 0$
You can slightly reduce the pain by simplifying before differentiating:

\displaystyle \begin{aligned}u = z^3 \bar{z}^5 + \bar{z}^3 z^5 &= (z\bar{z})^3(z^2+\bar{z}^2)\\ &= 2(x^2+y^2)^3(x^2-y^2) = 2(x^2+y^2)^2(x^4-y^4) = 2(x^8+2x^6y^2-2x^2y^6-y^8).\end{aligned}

Now all the tedious work has been done, and the differentiation is easy.

4. Originally Posted by Opalg
$\displaystyle 2(x^8+2x^6y^2-2x^2y^6-y^8)$

Now all the tedious work has been done, and the differentiation is easy.
Ah... It's like poetry!