What is Analytic functions? — Mathematics

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

$f$ is analytic at $x_0$ when it is the sum of a power series in $(x - x_0)$ 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 $x_0 = 0$ being Theorem 11.7). Analytic implies $C^\infty$ ; the converse fails: the flat function $\eu^{-1/x^2}$ (Exercise 11.7).

Examples

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

Expand $f(x) = \frac{1}{1-x}$ around $x_0 = \frac12$ : writing $x = \frac12 + h$ ,

\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 $\abs{2h} < 1$ , i.e. $\abs{x - \frac12} < \frac12$ . The new radius is exactly the distance from the new center to the singularity $x = 1$ : 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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