Pulling on a thread

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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 solution to Kepler’s equation in orbital mechanics. Thinking about Kepler’s equation lead to the posts on the Laplace limit and on series acceleration.

I may be done pulling on this thread. I don’t have anything else in mind that I want to explore for now, but you never know.

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John D. Cook, PhD

My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, applied math, and statistics.

Let’s talk. We look forward to exploring the opportunity to help your company too.