Complex Analysis/Radius of Convergence question.

Kemba Huskie
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Homework Statement


Question asks to show that if f is an entire function and bounded then it is polynomial of degree m or less.

Homework Equations


media%2Fa8d%2Fa8dc1d29-deb9-41c2-8537-b890cd541afc%2FphpscTgqj.png


The Attempt at a Solution


I tried plugging in the power series for f(z) and tried/know it is related to Liouville's Theorem somehow but I am just not gaining any traction. A picture of the question is attached below:
media%2Ff03%2Ff03104be-cddf-4510-8fee-46acd2526ce3%2Fphpx1dieD.png
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Make r big.
Your statement of the problem is wrong if you really mean the inequality of question 21 (in your attempt at a solution). A bounded entire function is a constant.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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