Analytic Functions: Is f(z) an Element of the Algebra of Polynominals?

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In summary, Timb00 is asking if an analytic function that can be locally expanded in a power series is a polynomial in z, and if this can be extended to the space of analytic functions. DonAntonio and Second DonAntonio confirm that the function is a polynomial only if the power series is finite, and Timb00 shares their concern about invariant theory and statistical physics. They ask for any ideas on how to extend the "pull back" mechanism from polynomials to analytic functions.
  • #1
timb00
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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
 
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  • #2
timb00 said:
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



Only if the power series (locally) is finite.

DonAntonio
 
  • #3
Second DonAntonio. A "polynomial", by definition, has a (finite) highest power.
 
  • #4
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.

Timb00
 

FAQ: Analytic Functions: Is f(z) an Element of the Algebra of Polynominals?

What are analytic functions?

Analytic functions are functions that are differentiable at every point in their domain. In other words, the derivative of an analytic function exists at every point in its domain. This property allows for the use of complex analysis to study these functions and make predictions about their behavior.

What is the Algebra of Polynomials?

The Algebra of Polynomials is a mathematical structure that consists of a set of polynomials and operations such as addition, subtraction, and multiplication. This algebraic structure is important in studying analytic functions because it allows us to manipulate and analyze these functions using algebraic techniques.

How do you determine if a function is analytic?

To determine if a function is analytic, we need to check if it is differentiable at every point in its domain. This can be done by taking the derivative of the function and checking for any points where the derivative does not exist.

What are some examples of analytic functions?

Some examples of analytic functions include polynomials, trigonometric functions, exponential functions, and logarithmic functions. These functions are differentiable at every point in their domains, making them suitable for complex analysis.

Why are analytic functions important in mathematics?

Analytic functions are important in mathematics because they have many useful properties that allow us to make predictions and solve problems. These functions can be expanded into power series, which can be used for approximations and calculations. Additionally, analytic functions have deep connections to other areas of mathematics, such as number theory and geometry.

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