Concept

Analytic function

A function that equals its own Taylor series on a neighbourhood of every point of its domain. Analytic implies infinitely differentiable and the converse fails on the real line, which is the whole difference between a flexible category and a rigid one: an analytic function vanishing on any interval vanishes everywhere.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

analysis · Taylor series

1/(1 + x²), expanded about 1.2. 1/(1 + x²) with Taylor sums of degree 2, 6, 14 about x = 1.2 rather than about zero. The interval they converge on reaches 1.562 either side of the centre.

The centre is a choice

A Taylor series is nearly always written about zero, and nothing about the construction prefers zero. Moving the centre moves the interval the series works on, and moving it repeatedly walks the function into places its first series could never reach.

analysis · Taylor series

11 points, equally spaced. 1/(1 + x²) and the polynomial of degree 10 through 11 of its points, spaced evenly across the interval. The worst error is 2.48e-1, at x = -2.350.

The points that ruin the fit

A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.

analysis · Taylor series

The coordinate change that makes the map a rotation. Graphs of the departure from the identity of the conjugacy between the circle map and the golden rotation, at three nonlinearities, each smooth and growing steeper with K.

How smooth the disguise is

An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.

dynamics · Mode locking

Named alongside it

The objects these essays reach for when they reach for this one.

Convergence Polynomial approximation Radius of convergence Taylor series Analytic continuation Approximation Circle map Complex numbers Conjugacy Continued fractions Derivative Diophantine approximation

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