Pulling on a thread

Often there’s a thread running through a sequence of my posts. Sometimes I make this explicit and sometimes I don’t. The latest thread started with this post commenting on a tweet that observed that exp(−x²) ≈ (1 + cos(sin(x + x))/2. Some people said online that that the approximation is simply due to the first few terms of the Taylor series on both sides matching up, so I wrote a follow up post explaining that it’s not that simple. The series for the left hand side alternates and converges very slowly, which lead to the post on naively summing an alternating series. The series for the right hand side lead to this point on partitions over permutations. Integrating the right hand side lead to this post on how the simplest numerical integration rule works shockingly well on some problems. The exact value of the integral turns out to be given by a Bessel function, details given in this post. Mr. Bessel’s interest in the functions now named after him started with looking closely at a…