Complex analysis - maximum modulus & analytic function

In summary, the conversation discusses two problems in complex analysis concerning the maximum modulus principle and analytic functions. The first problem involves determining whether the principle applies to an infinite strip, and the second problem involves showing that a bounded analytic function is constant. Hints are given, but the person asking for help has not shown any progress or understanding on the problems. They are urged to share their own work and difficulties in order for others to provide effective assistance.
  • #1
romiet3625
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[SOLVED] complex analysis - maximum modulus & analytic function

Hi all, I'm having difficulty figuring out how to do the following two problems in complex analysis. I need help!

1. Consider the infinite strip -[tex]\pi[/tex]< I am z < [tex]\pi[/tex]. Does maximum modulus principle apply to this strip? Why or why not? (Hint: e[tex]^{i\pi}[/tex] = e[tex]^{-i\pi}[/tex] = 1)



2. Show that if f(z) is analytic and Re f(z) is bounded in the complex plane, then f(z) is constant. What if I am f(z) is bounded? (Hint: Consider e[tex]^{f(z)}[/tex] and its absolute value)

Thank you for your time and thank you for any help.
 
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  • #2
You are given hints and you haven't used them? What have YOU done on this? No one can help you if you don't show us what YOU know about these problems and where you are having difficulties.
 

Related to Complex analysis - maximum modulus &amp; analytic function

1. What is the maximum modulus principle in complex analysis?

The maximum modulus principle states that the maximum modulus of an analytic function occurs on the boundary of its domain. In other words, if a function is analytic inside a certain region, the maximum value of its modulus will be on the boundary of that region.

2. How does the maximum modulus principle relate to the minimum modulus principle?

The minimum modulus principle is the opposite of the maximum modulus principle. It states that the minimum modulus of an analytic function occurs on the boundary of its domain. This means that the maximum and minimum values of the modulus of an analytic function are always on the boundary of its domain.

3. Can the maximum modulus principle be applied to functions with singularities?

No, the maximum modulus principle only applies to analytic functions. Functions with singularities, such as poles or essential singularities, do not satisfy the conditions for the maximum modulus principle to hold.

4. What is the significance of the maximum modulus principle in complex analysis?

The maximum modulus principle is a powerful tool in complex analysis. It is used to prove many important theorems, such as the open mapping theorem and the maximum modulus theorem. It also has applications in various areas of mathematics, such as in the study of harmonic functions and conformal mappings.

5. How is the maximum modulus principle used in practice?

In practice, the maximum modulus principle is used to find the maximum value of a given analytic function on a certain region. This can be useful in solving optimization problems or finding the behavior of a function on a specific domain. It can also be used to prove the existence of solutions to certain differential equations and to analyze the properties of complex functions.

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