Concept

Analytic continuation

Extending an analytic function beyond where its defining formula works. A new series is built at a point inside the current region and used where the old one has nothing to say, and the extension is unique wherever the region swept out has no hole - which is what lets the result be called the function rather than an extension of it.

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

p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated.

The size of a number with no formula

There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.

number · Partitions

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.

Convergence Analytic function Approximation Asymptotics Complex numbers Counting Estimate Generating function Growth rate Partition Pi Polynomial approximation

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