## Analytic Functions

A function
$f(x)$
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.