Find a harmonic conjugate $v$ so that $f=u+iv$ is holomorphic

Question

We have to show that, for $\xi \in \partial K(0,1)$, the function

$$ u_\xi = \frac{1-|z|^2}{|\xi - z|^2} $$

is harmonic in $K(0,1)$ and then find the harmonic conjugate to $u_\xi $.

It's very cumbersome to differentiate the function directly and prove it's harmonic that way, and I assume even more complicated to find its harmonic conjugate.

What about finding a function $v$ such that $f=u+iv$ is holomorphic? Something like (what it looks like for $\xi = 1$ )

$$f=\frac{g_1+i g_2}{1-z}$$

Any ideas?

Answer 1

You are on the right track. Consider the holomorphic function $$f(z):=\frac{\xi+z}{2(\xi-z)}$$ and show that $$\frac{1-|z|^2}{|\xi-z|^2}=\operatorname{Re}\left(f(z)\right).$$ Then the harmonic conjugate is simply $\operatorname{Im}\left(f(z)\right)$.