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

Adding phase-shifted sine waves | LTI systems

If two sine waves have the same frequency, but possibly different amplitudes and phases, their sum is another sine wave. How to find its amplitude and phase.

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Weierstrass approximation theorem: polynomials are dense in the space of functions continuous over a bounded interval.
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What exactly is modified about a modified Bessel function? johndcook.com/blog/2026/08/2…

What exactly is modified about a modified Bessel function?

What exactly is modified about modified Bessel functions? An explanation from pure math and applied math perspective. Other "modified" functions.

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Liouville's theorem: A bounded entire function is constant.
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Integral representations of means johndcook.com/blog/2024/01/0…

Integral representations of means | identric mean

Means of two numbers, a and b, may be written as f inverse of the average f over the interval [a, b]. Arithmetic, geometric, logarithmic, and identric means.

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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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