Here’s an interesting, and perhaps occasionally useful, identity: suppose you have a function defined on a two-dimensional space, \(f(\vec{r})\). And suppose that this function is independent of the angle of \(\vec{r}\) but rather is only a function of the magnitude \(r = \norm{\vec{r}}\). Then

(where \(\ln r\) is shorthand for \(\ln\frac{r}{r_0}\) where \(r_0\) is some constant) In other words, the radial term of the polar Laplacian is equal to the second logarithmic derivative divided by \(r^2\) for rotationally invariant functions.

At this point you might be wondering, why the heck is this interesting? Well, for me, it’s because my current research project happens to involve logarithmic derivatives of rotationally invariant functions in 2D space. But for (almost) everyone else, this is related to the way electric potential drops off with distance in 2D space.

Consider, for example, a wire that carries a charge density \(\lambda\) (but no current). Since this system is translationally symmetric (nothing changes if you …