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Mathematics · Glossary

What is Analytic functions?

Also known as: analytic function

Definition 11.13 University Mathematics — Year 2 · Chapter 11 — Power Series

ff is analytic at x0x_0 when it is the sum of a power series in (x−x0)(x - x_0) on a neighborhood; on an interval, when at every point. Sums of power series are analytic inside their disk (rearrangement of the expansion — admitted at this level for re-centering, the case x0=0x_0 = 0 being Theorem 11.7). Analytic implies C∞C^\infty; the converse fails: the flat function e−1/x2\eu^{-1/x^2} (Exercise 11.7).

Examples

Example 11.14 (Re-centering, and the radius as a distance)

Expand f(x)=11−xf(x) = \frac{1}{1-x} around x0=12x_0 = \frac12: writing x=12+hx = \frac12 + h,

11−x=112−h=21−2h=∑n≥02n+1 hn=∑n≥02n+1(x−12) ⁣n,\frac{1}{1 - x} = \frac{1}{\frac12 - h} = \frac{2}{1 - 2h} = \sum_{n\geq0} 2^{n+1}\,h^n = \sum_{n\geq0} 2^{n+1}\Bigl(x - \frac12\Bigr)^{\!n},

valid for ∣2h∣<1\abs{2h} < 1, i.e. ∣x−12∣<12\abs{x - \frac12} < \frac12. The new radius is exactly the distance from the new center to the singularity x=1x = 1: re-centering shrinks (or grows) the disk to fit the nearest obstruction. Closing insight: this is the picture behind the definition of analyticity — one function, many local power series, each living on the largest disk avoiding the trouble; the Year 3 volume turns the heuristic “radius == distance to the nearest complex singularity” into a theorem.

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