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?