# Holomorphic function reduces to a polynomial

• snipez90
In summary, the conversation discusses finding a way to show that a holomorphic function with a given growth condition is a polynomial. The suggested approach involves considering the reciprocal of the function and using the maximum modulus principle. Liouville's theorem may also be relevant.
snipez90

## Homework Statement

Let f: C -> C be a holomorphic function such that there is a constant R such that |z| >
R implies |f(z)| > R. Show that f is a polynomial.

## Homework Equations

Not sure, I pulled this randomly from a complex analysis qualifying exam.

## The Attempt at a Solution

So from experience a typical way to show that a holomorphic function is a polynomial is to apply Cauchy estimates (e.g. the immediate estimates from the Cauchy integral formula). However that approach doesn't seem to work here, since we usually have to let the boundary circle in the Cauchy integral formula either get larger and larger or smaller and smaller. To me it's not clear how the given growth condition gives estimates.

I've also thought about the maximum modulus principle, but I don't how to use it well, even if it does apply here. Can someone provide a hint? Thanks in advance.

You might want to look at 1/f(z). It's bounded, except for the possibility of poles inside |z|<R. Try to make a new holomorphic function from this

All right thanks. I did consider 1/f, but erroneously thought of Liouville. I'll try your suggestion.

Well, Liouville will come into play. But first you need to find a slightly different function that's actually bounded

## 1. What is a holomorphic function?

A holomorphic function is a complex-valued function that is differentiable at every point in its domain. It is also known as an analytic function.

## 2. How does a holomorphic function reduce to a polynomial?

A holomorphic function reduces to a polynomial if it is defined on a simply connected domain and satisfies Cauchy's integral formula. This means that the function can be expressed as a power series with only finitely many non-zero terms.

## 3. What are the advantages of reducing a holomorphic function to a polynomial?

Reducing a holomorphic function to a polynomial allows for simpler and more efficient calculations, as polynomials are easier to manipulate and evaluate compared to general holomorphic functions.

## 4. Are all holomorphic functions reducible to polynomials?

No, not all holomorphic functions can be reduced to polynomials. Some functions, such as exponential and trigonometric functions, cannot be expressed as polynomials.

## 5. Can a non-holomorphic function be reduced to a polynomial?

No, only holomorphic functions can be reduced to polynomials. Non-holomorphic functions, such as functions with singularities, cannot be expressed as polynomials.

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