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Analysis Fact @AnalysisFact

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

Joined December 2009
  • Tweets 12,898
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  • Followers 134,362
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Analysis Fact@AnalysisFact
3 Nov 2022
Applied mathematics consulting johndcook.com/blog/applied-m…
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'... all exact science is dominated by the concept of approximation.' -- Bertrand Shaw
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'Young men should prove theorems, old men should write books.' -- G. H. Hardy
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Sine approximations for small angles johndcook.com/blog/2010/07/2…

Sine approximation for small angles

The statement "sin x is approximately x for small x" means more than just saying small angles have small sines. It means the error is small EVEN compared to x.

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''Obvious' is the most dangerous word in mathematics.' -- E. T. Bell
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Diagram of special function relationships johndcook.com/blog/2010/09/2…

Diagram of special function relationships

Here's a first pass at a diagram showing the relationships between special functions that commonly come up in applications. These notes give a larger image and explain what everything on the diagram...

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'A profound study of nature is the most fertile source of mathematical discoveries.' -- Joseph Fourier
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Analysis Fact retweeted
Algebra Etc.@AlgebraFact
Jun 10
Pulling on a thread: How an approximation lead to a series of nine posts. johndcook.com/blog/2026/06/1…

Pulling on a thread

The thread running through the latest posts.

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'Beauty is the first test: there is no permanent place in the world for ugly mathematics.' -- G. H. Hardy
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Trapezoid rule and Romberg integration johndcook.com/blog/2020/02/1…

Trapezoid rule and Romberg integration

Here we look at the simple trapezoid rule and Romberg's clever extension of the trapezoid rule. We will also demonstrate the value of memoization.

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'It is not the answer that enlightens, but the question.' -- Eugene Decouvertes
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Complex exponentials. Where people often go wrong. johndcook.com/blog/2018/11/3…

Complex exponential function | how people go wrong

Most of the rules from high school algebra carry over to complex numbers, with one important exception: exponentials.

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New post: Aitken acceleration before Aitken johndcook.com/blog/2026/06/0…

Aitken acceleration before Aitken

Peter's method for solving Kepler's equation was a special case of Aitken acceleration that came 35 years later.

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New post: The Laplace limit johndcook.com/blog/2026/06/0…

The Laplace limit for solving Kepler's equation

The power series solution to Kepler's equation has a radius of convergence of around 0.6627 for no obvious reason.

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From Kepler to Bessel How Kepler's equation led to the definition of Bessel functions johndcook.com/blog/2026/06/0…

How Kepler's equation led to Bessel functions

Solving Kepler's equation is a common task in astronomy and orbital mechanics. Solving this equation with Fourier series leads to Bessel functions.

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Every continuous function of many variables is a finite composition of continuous functions of two variables. -- Kolmogorov
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Three views of differential equations johndcook.com/blog/2011/01/0…

Differential equations: three views

The most common view of differential equations may be sheer terror, but those who get past terror may have one of the following perspectives. Naive view: All differential equations can be solved in...

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Evaluating functions of quaternion argument tpfto.wordpress.com/2020/05/…

Evaluating functions of quaternion argument

A recent Mathematica Stack Exchange thread got me thinking again about the evaluation of an arbitrary analytic function $latex f(z)$ of quaternion argument. Working through the derivation resulted …

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Mr. Bessel’s eponymous functions johndcook.com/blog/2026/06/0…
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Three body problem fractal mathstodon.xyz/@noneuclidean…

Non-Euclidean Dreamer (@noneuclideandreamer@mathstodon.xyz)

Attached: 1 image 3-Body-problem Fractal: Planet with initial position (1|0|t) and initial velocity (x|y|0) moves in a binary star system. Red and Green tell you how close the planet gets to either...

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'Mathematics is the science that uses easy words for hard ideas.' -- Edward Kasner
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