Applied mathematics consulting johndcook.com/blog/applied-m…

The Bessel functions J_n satisfy J_{n-1}(z) + J_{n+1}(z) = 2 J_n(z)/ z

The integral of an odd function over an interval (-a, a) is zero.

Analytic functions are infinitely differentiable but the converse is not true. For example, exp(-1/x²) at 0.

What << and >> mean in context johndcook.com/blog/2021/09/1…

Γ(z) Γ(1-z) = π / sin(πz)

Airy's equation: y'' - xy = 0. Solutions are oscillatory for negative x, monotonic for positive x.

Adding phase-shifted sine waves johndcook.com/blog/2020/08/1…

Weierstrass approximation theorem: polynomials are dense in the space of functions continuous over a bounded interval.

What exactly is modified about a modified Bessel function? johndcook.com/blog/2026/08/2…

Liouville's theorem: A bounded entire function is constant.

Integral representations of means johndcook.com/blog/2024/01/0…

Γ(1/2) = √π

A Cantor set is uncountable and yet can have Lebesgue measure zero (or positive measure).

The rational numbers have Lebesgue measure zero.

To integrate the impossible integral johndcook.com/blog/2020/05/1…

Monoliths and Lagrange points A thread from last night, edited into a blog post, with additions and corrections. johndcook.com/blog/2026/10/1…

Analytic functions dilate and rotate tiny disks, but can't flip them over. Complex conjugate is not analytic because it flips disks over.

Strange attractors and Imagery uh.edu/engines/epi2599.htm

Interpolation is generally more trustworthy than extrapolation.

How to start with a minimal math library and bootstrap it to compute the functions not included johndcook.com/blog/2021/01/0…