@AnalysisFact

Daily tweets about real and complex analysis and related topics. From @JohnDCook.

johndcook.com/blog/applied-math/
Joined December 2009
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Applied mathematics consulting johndcook.com/blog/applied-m…
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Γ(1/2) = √π
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A Cantor set is uncountable and yet can have Lebesgue measure zero (or positive measure).
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The rational numbers have Lebesgue measure zero.
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To integrate the impossible integral johndcook.com/blog/2020/05/1…

Integrate square root of sine

An "impossible" integral creates an opportunity to discuss in what sense it is impossible and in what sense it is possible.

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Monoliths and Lagrange points A thread from last night, edited into a blog post, with additions and corrections. johndcook.com/blog/2026/10/1…
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Analytic functions dilate and rotate tiny disks, but can't flip them over. Complex conjugate is not analytic because it flips disks over.
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Strange attractors and Imagery uh.edu/engines/epi2599.htm
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Interpolation is generally more trustworthy than extrapolation.
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How to start with a minimal math library and bootstrap it to compute the functions not included johndcook.com/blog/2021/01/0…

Bootstrapping a small math library

How to start with a minimal math library and compute functions that are not included. e.g. how to use arc tangent to compute the other inverse trig functions.

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1/Γ(z) is an entire function, i.e. it is analytic everywhere in the complex plane.
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Γ(z+1) = z Γ(z) for complex z. Γ(n+1) = n! if n is a non-negative integer.
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Keyhole contours johndcook.com/blog/2022/07/1…

Keyhole contour integrals

Keyhole contours are often used to evaluate real integrals. Also known as a Hankel contour.

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'I have made a living off the maximum principle.' -- Louis Nirenberg
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Analysis Fact retweeted
Matheorems
@matheoremsOct 9
The computer that landed Apollo 11 on the Moon computed sine with a polynomial of just three terms. sin(πx/2) ≈ 2((C₅x² + C₃)x² + C₁)x, used only for |x| ≤ 1. The constants are not Taylor coefficients. Taylor's C₃ would be −0.3229820, the AGC stored −0.3216147, nudged so the error wiggles across the interval instead of piling up at the end. Result: max error 1.1×10⁻⁴ versus 4.5×10⁻³ for Taylor, about 40 times better at the same cost. Good engineering often means choosing better constants.
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Unified Pythagorean theorem johndcook.com/blog/2022/08/2…

Unified Pythagorean Theorem

A generalization of the Pythagorean theorem that holds in the plane, on a sphere, and on a pseudosphere.

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'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold
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The uniform limit of continuous functions is continuous. The pointwise limit might not be continuous.
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Sum and mean inequalities move in opposite directions johndcook.com/blog/2020/05/1…

Sum and mean inequalities move in opposite directions

When you look at means and sums with varying exponents, as with RMS or Lebesgue norms, the inequalities for means and sums go in opposite directions.

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d(dw) = 0 for any differential form w.
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The curl of a gradient is zero. The divergence of a curl is zero.
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