## Analytic Functions

\[f(x)\]

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.