Q:

Invariance of the cylindrical Laplace equation under conformal transform

hello,
it is often said that a conformal mapping preserves the Laplace equetion in 2D.
However, if this is true for the cartesian coordinates (x,y), where the laplacian is:
$$
\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2}=0
$$
is it true for the cylindrical coordinates (r,z) where:
$$
\frac{\partial^2 \phi}{\partial r^2}+\frac{\partial^2 \phi}{\partial z^2}+\frac{1}{r}\frac{\partial \phi}{\partial r}=0?
$$
More precisely, if you have a function $\phi(r,z)$ verifying the previous equation, is it also verified by $\psi(u,v)$ is $u+i v=f(z+ir)$ and f is analytic/holomorphic?

A:

No. Take $\phi=r^2-2z^2$ and $f(z+ir)=(z+ir)^2$, thus $u=r^2-z^2$, $v=2rz$. Then $\phi(u,v)=r^4+z^4-10r^2z^2$ is not cylindrical-harmonic.

