Harmonic conjugate of $u(r,\theta)=r^3\cos(⁡3\theta)$

Question

Verify that the following function $u$ are harmonic, and in each case give a conjugate harmonic function $v$ (i.e $v$ such that $u+iv$ is analytic)

$$u(r,\theta)=r^3\cos⁡(3\theta)$$

I need to find the harmonic conjugate of the equation but I can't seem to prove it is harmonic. Does anybody got any idea? Thank you so much.

Answer 1

By de Moivre's theorem $$r^3\cos3\theta=\operatorname{Re}(r^3(\cos3\theta+i\sin3\theta))=\operatorname{Re}(r^3(\cos\theta+i\sin\theta)^3)=\operatorname{Re}((re^{i\theta})^3)=\operatorname{Re}(z^3)$$ where $z$ is an arbitrary complex number. Thus $r^3\cos3\theta$ is harmonic and its harmonic conjugate is $r^3\sin3\theta$.

The surface obtained as $z=r^3\cos3\theta$ is called the monkey saddle.

Answer 2

$z^3$ is holomorphic, hence $\Re(z^3)$ and $\Im(z^3)$ are harmonic conjugates; writing

$z = re^{i\theta}, \tag 1$

we see that

$z^3 = (re^{i\theta})^3 = r^3(e^{i\theta})^3 = r^3(\cos 3\theta + i \sin 3 \theta) =r^3\cos 3\theta + ir^3 \sin 3 \theta); \tag 2$

in follows that

$\Re(z^3) = r^3\cos 3\theta \tag 3$

and

$\Im(z^3) = r^3\sin 3\theta, \tag 4$

are both harmonic functions, with $r^3\sin 3\theta$ conjugate to $r^3\cos 3\theta$.