Concept

Singularity

A point where a function fails to be analytic - a pole, a branch point, or something worse. Its importance is that it is never local news: the distance from an expansion point to the nearest singularity is exactly the radius on which that expansion converges, however far away and however invisible it is.

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

The equation the objects satisfy. A diagram of the decomposition C = 1 + xC², with a table of the first several coefficients computed two ways: by the convolution the equation prescribes, and from the closed form.

The equation a sequence satisfies

Write the whole sequence as the coefficients of one series, and the recursion becomes an equation with a square in it. Solving the equation by the ordinary quadratic formula produces the closed form, the growth rate and the correction term, none of which the recursion offers.

discrete · Catalan numbers

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

Named alongside it

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

Analytic continuation Analytic function Asymptotics Binomial coefficient Catalan numbers Complex numbers Convergence Convolution Generating function Polynomial approximation Power series Radius of convergence

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