Proof of Liouville's Theorem +

In summary, the conversation covers Rouche's Theorem and Liouville's Theorem, using the maximum modulus theorem to find the max and min of a function over a unit disk, and finding a Mobius transformation with specific given values.
  • #1
MurraySt
8
0
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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  • #2
Is this homework? As if it is it's in the wrong forum.
 

What is Liouville's theorem?

Liouville's theorem is a mathematical concept that states that if a function is both continuous and bounded, then it must be a constant function. It is often used in complex analysis and differential equations.

What is the "proof" of Liouville's theorem?

The proof of Liouville's theorem involves using the Cauchy integral formula, which states that the value of a function at a point is equal to the average value of the function over a closed contour surrounding that point. By applying this formula to a function that is both continuous and bounded, we can show that the function must be constant.

Why is Liouville's theorem important?

Liouville's theorem is important because it helps us to understand and analyze complex functions. It can also be used to prove other theorems in mathematics, such as the fundamental theorem of algebra.

How is Liouville's theorem related to the concept of holomorphic functions?

Liouville's theorem is closely related to holomorphic functions, which are complex-valued functions that are differentiable at every point in their domain. This is because Liouville's theorem is often used to prove the identity theorem for holomorphic functions, which states that if two holomorphic functions are equal on a set with an accumulation point, then they are equal everywhere in their domain.

Are there any exceptions to Liouville's theorem?

Yes, there are some exceptions to Liouville's theorem. For example, if a function is continuous and bounded on the complex plane, but has an essential singularity at a point, then it is not a constant function. In general, Liouville's theorem only applies to entire functions, which are complex-valued functions that are defined and holomorphic on the entire complex plane.

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