How Do I Use Rouche's Theorem to Prove Liouville's Theorem?

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SUMMARY

This discussion focuses on using Rouche's Theorem to prove Liouville's Theorem, specifically through the analysis of the function f(z) = z^2 - 2 within the unit disk. The maximum modulus theorem is relevant for finding the maximum and minimum of |f(z)|. Additionally, the set defined by e^(1/z) for 0 < |z| < epsilon is discussed, noting that it covers the entire complex plane except for the point at zero. The conversation also touches on finding a Möbius transformation given specific function values, emphasizing the use of the cross ratio.

PREREQUISITES
  • Understanding of Rouche's Theorem
  • Familiarity with Liouville's Theorem
  • Knowledge of the maximum modulus theorem
  • Concept of Möbius transformations and cross ratios
NEXT STEPS
  • Study the application of Rouche's Theorem in complex analysis
  • Research the implications of Liouville's Theorem in the context of entire functions
  • Learn how to apply the maximum modulus theorem to complex functions
  • Explore the properties and applications of Möbius transformations
USEFUL FOR

Students of complex analysis, mathematicians interested in function theory, and anyone seeking to understand the relationship between Rouche's Theorem and Liouville's Theorem.

MurraySt
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I'm asked to use Rouche's Theorem to prove Liouville's - I really don't have much of a clue as proofs are not my strong suit.
Next up: Find the max and min of abs(f(z)) over the unit disk where f(z) = z^2 - 2

Do I use the maximum modulus theorem?Lastly I'm given epsilon>0 and the set e^(1/z) where 0<abs(z)<epsilon. This set is equal to the entire complex plane minus 0 as e^(1/z) cannot take on that value. The question is: What can I say about the set? Besides the fact that it cannot be 0 I'm out of ideas.Thanks as always

Edit: One final question
I'm give that f(1) = 1, f(-1) = i and f(-i) = 1. I need to find a Mobius transformation.I believe I need to use the cross ratio - but the problem is that Mobius transformations should send something to 0, 1 and infinity (which this one does not) how can I get around this issue?
 
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Is this homework? As if it is it's in the wrong forum.
 

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