Analysis Fact

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

exp(iz) = cos(z) + i sin(z)

When does a function satisfy an addition theorem? johndcook.com/blog/2023/07/1…

For z in the right half-plane |Γ(z)| ≤ |Γ(Re z)|.

Stirling's approximation for factorial en.wikipedia.org/wiki/Stirling%…

The dual space of a Banach space X is the space X^* of continuous linear functions from X to R.

More here: johndcook.com/blog/2026/03/1…

If f is holomorphic in the right half-plane and 1. f(z + 1) = z f(z) 2. f(1) = 1 3. f(z) is bounded for {z: 1 ≤ Re z ≤ 2} then f is the gamma function.

Roughly speaking, the Riemann integral chops up the domain but the Lebesgue integral chops up the range.

The space of continuous linear maps between two Banach spaces is itself a Banach space.

Tighter bounds on the remainder in an alternating series johndcook.com/blog/2026/03/1…

An everywhere chaotic map on the sphere johndcook.com/blog/2021/10/1…

A topological space is locally compact if every point is contained in an open set whose closure is compact.

If a function f: [a, b] -> R is absolutely continuous, f has bounded variation.

Milman's theorem: A uniformly convex Banach space is reflexive.

Why Mathematica does not simplify sinh(arccosh(x)) johndcook.com/blog/2026/03/1…

The paradox of derivatives and integrals statmodeling.stat.columbia.edu/2026/03/14/the…

sinh(z) = -i sin(iz)

Computing Fourier series coefficients with the FFT johndcook.com/blog/2021/03/2…

“The key to understanding the Dirac delta function is that it’s not a function on real numbers but a continuous linear functional on the space of smooth functions with compact support.”

A linear map between Banach spaces is continuous iff the image of the unit ball is bounded.