# 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?

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How about $$\frac{\partial^2 \phi}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2 \phi}{\partial z^2}+\frac{1}{r}\frac{\partial \phi}{\partial r}=0$$ –  Gerald Edgar Oct 13 '10 at 15:15
Maybe the change of variables you want is $ue^{iv} = f(re^{iz})$ ... –  Gerald Edgar Oct 13 '10 at 15:27
To elaborate on Gerald's comment: the coordinate change you wrote down is not a conformal mapping [see en.wikipedia.org/wiki/Conformal_map ] in general. Also, this question has nothing to do with conformal field theory. –  Willie Wong Oct 13 '10 at 15:41
but that is suited when you want to apply you conformal mapping to the $(r,\theta)$ plane and not the $(r,z)$ plane, or? –  Mermoz Oct 13 '10 at 15:50
wait a second, what do you actually mean by the cylindrical coordinates in 2D? Are you actually interested in functions defined over 3 dimensional space but is constant along the azimuthal direction? –  Willie Wong Oct 13 '10 at 18:30

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.