Relationship between exp(1/z) and the magnetic field I will preface this by saying that I'm a math student, so I know a fair bit of math but almost no physics beyond what is learned in a first year undergraduate course.
I was recently looking at this article on the Wiki, and I noticed that the graph of $\exp{\left(\frac{1}{z}\right)}$ looks incredibly similar to the diagrams I've seen of the magnetic field around a dipole. This connection can be made more explicit:
Let $f:\mathbb{R}^3\to\mathbb{C}$ be defined by $(x,y,z)\mapsto\exp{\left(\frac{1}{x+i\sqrt{y^2+z^2}}\right)}$.
Then, if one considers an infinitely small magnetic dipole at $(0,0,0)$ with north facing towards $-x$, the direction of the magnetic field at $(x,y,z)$ seems to be given by $\nabla |f(x,y,z)|$, and the magnitude seems to be given by the phase of $f(x,y,z)$ (normally this will only give the magnitude modulo $2\pi$, but if one looks at $f$ as mapping onto a Riemann surface, it should give the full magnitude).
Is there a reason for this similarity? One often sees exponentials in the solutions to differential equations, so I could imagine there is some differential equation governing magnetism for which $f$ is a solution. Or is there perhaps some other explanation?
 A: These fields look superficially similar, but they're not the same.
The claim seems to be equivalent to saying that the magnetic field lines of a dipole are the same as the curves of constant phase for $\exp(1/z)$. In the latter case, that means $\text{Im}(1/z)$ is constant, which implies that
$$ b \propto |z|^2$$
where $z = a+bi$. Meanwhile, in the magnetic dipole case, it is well known that the field lines in the plane satisfy
$$r \propto \sin^2 \theta$$
which implies that
$$y \propto r^{3/2}.$$
This is not the same as your proposed function, which is more like $y \propto r^2$. They only look similar because they're both $y \propto r^n$ for some $n > 1$. 
Sometimes, it is useful to represent problems in physics using complex functions, because the components of holomorphic functions are harmonic. However, this only works in two dimensions (or more generally an even number), while the magnetic dipole field is inherently three-dimensional. Your idea might apply to a "two-dimensional" dipole, e.g. a dipole formed by two infinite oppositely-charged lines, though.
