Analysis Fact

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

A real-valued function on a Banach space is lower semi-continuous iff it is closed.

Introduction to the Legendre transform jmanton.wordpress.com/2010/11/21/int…

Mathematics. "Guitar Pick Equation." By John D. Cook, @JohnDCook, Used with permission, johndcook.com/blog/2026/05/0…

Gauss proved the Fundamental Theorem of Algebra in his doctoral dissertation in 1799.

A simple proof that there is no measure on infinite dimensional Hilbert space similar to the standard Lebesgue measure on Euclidean space ℝⁿ. #math #calculus #MeasureTheory #geometry #HilbertSpace

A convex function on an open interval is differentiable except possibly on a countable set of points.

Stable and unstable recurrence relationships johndcook.com/blog/2020/01/1…

Approximating even functions by powers of cosine johndcook.com/blog/2026/04/3…

A compact convex set in finite dimensions is the convex hull of its extreme points.

log(1 + x) is approximately x for small values of x.

Three kinds of generalized derivative, illustrated by the ramp function. johndcook.com/blog/2026/04/3…

The product of the mth and nth Chebyshev polynomials is the average of the (m + n)th and (m – n)th Chebyshev polynomials.

Sinc approximation johndcook.com/blog/2020/06/0…

Turning a trick into a technique johndcook.com/blog/2026/04/2…

Leibniz showed that π/4 = 1 - 1/3 + 1/5 - 1/7 ... in 1674.

A set is convex iff its indicator function is a convex function. (Indicator function is infinite off the set.)

A function is convex iff its epigraph is a convex set.

A midpoint convex function is not necessarily convex, but a strictly midpoint convex function is strictly convex.

For the generalization to nth derivatives, see this post. johndcook.com/blog/2026/04/2…

The generalization isn't obvious. Here's the second derivative.

Quotient rule in a form that suggests a way to generalize to repeated derivatives