\[f(x)\]
is analytic on an interval \[I\]
(which may be open or closed) if it can be differentiated indefinitely for every \[x \in I\]
i.e. if \[\frac{d^n( f(x))}{dx^n}\]
exists for all \[n\]
.If a function is analytic on an interval it can be expanded in a Taylor series around every interior point
\[x_0\]
of that interval, so we can write \[f(x)= \sum_k \frac{1}{k!} \frac{d^k (f(x))}{dx^k} |_{x=x_0} (x-x_0)^k\]
.Analytic functions need not be analytic everywhere.
\[f(x) = \frac{1}{x}\]
is analytic except at \[x=0\]
. All the trigonometric functions, expoentials, logarithms, polynomials, roots and reciprocals and many products, sum, quotients and compositions are analytic on some interval.