Complex analysis - maximum modulus & analytic function

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SUMMARY

The discussion focuses on two problems in complex analysis regarding the maximum modulus principle and properties of analytic functions. The first problem examines whether the maximum modulus principle applies to the infinite strip defined by -π < Im(z) < π, concluding that it does not apply due to the nature of the strip. The second problem asserts that if f(z) is analytic and Re(f(z)) is bounded in the complex plane, then f(z) must be constant, while also exploring the implications if f(z) itself is bounded.

PREREQUISITES
  • Understanding of complex analysis concepts, particularly the maximum modulus principle.
  • Familiarity with analytic functions and their properties.
  • Knowledge of the implications of bounded real parts of complex functions.
  • Experience with exponential functions in the context of complex variables.
NEXT STEPS
  • Study the maximum modulus principle in detail, including its exceptions.
  • Explore the properties of analytic functions and the implications of Liouville's theorem.
  • Research the behavior of complex functions under bounded conditions, particularly Re(f(z)).
  • Investigate the role of exponential functions in complex analysis, focusing on their absolute values.
USEFUL FOR

Students and professionals in mathematics, particularly those specializing in complex analysis, as well as educators seeking to deepen their understanding of analytic functions and their properties.

romiet3625
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[SOLVED] complex analysis - maximum modulus &amp; 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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