Anthony Bonfils at the blackboard.
Nordita Reflections is a series where researchers share small moments from their work: ideas, thoughts and discoveries along the way.
In the first reflection, postdoc Anthony Bonfils writes about a deceptively simple equation.
On the solutions of tan(x) = x
A transcendental equation that often appears in physics is:
tan(x) = x.
We learn early on that this deceptively simple equation cannot be solved analytically, so the instinct is simply to let a computer find the solutions. Few people try to approach it differently.
By chance, I stumbled upon a very good numerical approximation for the n-th root of this equation:
xₙ ≈ π √(n+n²)
It works remarkably well for large n. This made me wonder whether it might be possible to construct a proper asymptotic expansion using 1/n as a small parameter.
For a while I fulfilled the parodic definition of an applied mathematician: covering pages with calculations, only to throw them in the bin at the end of the day because they were all wrong.
Eventually, I found the trick and obtained a clean asymptotic series for xₙ. I was delighted, and immediately started thinking about applying the method to more complicated cases, such as the zeros of Bessel functions.
Then I discovered that it had already been done by Stokes in 1856. As professor John Wettlaufer used to say, there is no statute of limitations in the literature.
In the end, I was only about 170 years late.
