Is f(z) = e^(z^2) for all z in C?

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SUMMARY

The discussion centers on proving that the function f(z) = e^(z^2) for all z in C, given the conditions |f(z)| >= 1/3|e^(z^2)| for all z in C, f(0) = 1, and that f(z) is entire. The approach involves defining g(z) = e^(z^2) and analyzing the function h(z) = g(z)/f(z). By demonstrating that h(z) is entire and has a bounded modulus, the conclusion follows that f(z) must equal g(z) across the complex plane.

PREREQUISITES
  • Understanding of entire functions in complex analysis
  • Familiarity with the properties of bounded functions
  • Knowledge of the maximum modulus principle
  • Experience with complex function theory, particularly regarding limits and continuity
NEXT STEPS
  • Study the maximum modulus principle in complex analysis
  • Learn about Liouville's theorem and its implications for entire functions
  • Explore the properties of bounded entire functions
  • Investigate the implications of the Cauchy-Riemann equations on entire functions
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, particularly those specializing in complex analysis, as well as researchers exploring properties of entire functions and their applications.

Matt100
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Assume |f(z)| >= 1/3|e^(z^2)| for all z in C and that f(0) = 1 and that f(z) is entire. Prove that f(z) = e^(z^2) for all z in C.

How do you start for this.
 
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Let g(z) = e^(z^2) and consider the function g(z)/f(z). Show that it is entire and has a bounded modulous. You should be able to continue from there.
 

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