^^ [[Schutz - 1 - Some Basic Mathematics|1. Some Basic Mathematics]] << [[Schutz - 1.2 - Mappings|1.2 Mappings]] | [[Schutz - 1.4 - Group Theory|1.4 Group Theory]] >> ^60ba52 # Real Analysis > **Analytic Functions** > A real function of a single real variable, $f(x)$ is said to be *analytic* at $x = x_0$ if it has a Taylor expansion about $x_0$ that converges to $f(x)$ in some neighborhood of $x_0$: > $f(x) = f(x_0) + (x-x_0)\left(\frac{df}{dx}\right) + \frac{1}{2}(x-x_0)\left(\frac{d^2f}{dx^2}\right)+ \ldots$ > Naturally, a function needs to be infinitely [[Schutz - 1.2 - Mappings#^319309|differentiable]] at $x = x_0$ for it to be analytic, but not all infinitely differentiable functions are analytic. Analytic functions are denoted by $C^\omega$. ^e9e4e3 > **Square-integrable functions** > A function $g(x_1, x_2, \ldots, x_n)$ defined on an [[Schutz - 1.1 - The Space R^n and its Topology#^224639|open region]] $S$ of $\mathbb{R}^n$ is called *square-integrable* if the multiple integral > $ \int_S [g(x_1, x_2, \ldots x_n)]^2 dx_1 dx_2 \ldots dx_2 $ > exists. > It is a theorem of complex analysis that any square-integrable function $g$ may be approximated by an [[Schutz - 1.3 - Real Analysis#^e9e4e3|analytic function]] $g'$ in such a way that the integral of $(g - g')$ over $S$ may be made as small as one wishes. ^4bdffa