# Analytic functions

1. May 29, 2012

### timb00

Hey everybody,

I have a short question. Suppose you have an analytic function f(z) such that you can, at least locally, expand it in a power series. Is this function then (locally) an element of the algebra of polynominals in z?

I hope I formulate the question in the right way.

Best wishes, Timb00

2. May 29, 2012

### DonAntonio

Only if the power series (locally) is finite.

DonAntonio

3. May 29, 2012

### HallsofIvy

Staff Emeritus
Second DonAntonio. A "polynomial", by definition, has a (finite) highest power.

4. May 30, 2012

### timb00

Thank you for your fast answers. They have confirmed my expectancy. I ask this question because I am concerned with invariant theory and statistical physics. I learned that every polynominal function that is invariant under the action of a Lie group is a pull back of a function on some matrix space depending on the Lie group. Due to this observation I ask my self how this is extended to the space of analytic functions.

Maybe some one of you has an Idea how to extend this "pull back" mechanism to analytic functions.

Best, Timb00