Analysis Fact

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

If the Fourier transform of f(x) is F(t), the transform of f(x/m) is m F(mt).

In p-adic numbers, if a sequence converges, it converges unconditionally (but not necessarily absolutely).

The real and imaginary parts of an analytic function are harmonic functions.

A more powerful alternating series theorem johndcook.com/blog/2021/11/0…

A conditionally convergent series can be re-arranged to converge to any value.

Once you learn calculus, you've caught up to what was the cutting edge of math in 1687.

Rate of convergence in Newton's method johndcook.com/blog/2021/01/1…

Newton's method and r-means johndcook.com/blog/2021/04/0…

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