Analysis Fact

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

[Analysis Fact @AnalysisFact]

Applied mathematics consulting

https://t.co/AwpFbcBvIn

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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?

https://t.co/0GhlD9UkO5

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Liouville's theorem: A bounded entire function is constant.

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Integral representations of means https://t.co/kIdsD7iiYT

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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 https://t.co/vgpyCyyYEE

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Monoliths and Lagrange points

A thread from last night, edited into a blog post, with additions and corrections.

https://t.co/w2kOdGGGcJ

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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 https://t.co/wTSdi4JLzX

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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 https://t.co/vvTEyHm0qG

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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 https://t.co/cCtWfm9K1j

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'I have made a living off the maximum principle.' -- Louis Nirenberg

[Matheorems @matheorems]

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 https://t.co/1g8q2Ha1kl

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'Mathematics is the part of physics where experiments are cheap.' -- V. I. Arnold