抽丝剥茧Pulling on a thread
数学探索往往像抽丝剥茧，一系列看似无关的帖子背后可能隐藏着连贯的逻辑。最新的探索线索始于一个关于 exp(−x²) ≈ (1 + cos(sin(x) + x))/2 的近似公式。作者深入挖掘了该公式的数学原理，反驳了网上认为这只是简单的一阶近似的观点。通过追踪这个线索，揭示了这个看似巧合的近似值背后深度的数学机制。
John
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.
需要完整排版与评论请前往来源站点阅读。