Analytic Functions

A function  
  is analytic on an interval  
  (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  
If a function is analytic on an interval it can be expanded in a Taylor series around every interior point  
  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  
. All the trigonometric functions, expoentials, logarithms, polynomials, roots and reciprocals and many products, sum, quotients and compositions are analytic on some interval.

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