Analytic function
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
Where to take the samples
Every numerical integral is a weighted sum of samples of the curve. Spacing the samples evenly gives errors that shrink like a power of their number; choosing where they go, as Gauss did in 1814, makes six samples exact for every polynomial up to degree eleven and turns the power into a geometric rate. The surprise is that the humble trapezoid rule, evenly spaced and exact only for straight lines, does the same thing on any smooth curve that goes once round a circle — and that a single corner defeats every one of them.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvergencePolynomial approximationRadius of convergenceTaylor seriesFourier seriesInterpolationSingularityAliasingAnalytic continuationApproximationChebyshev polynomialCircle map