Showing Uniqueness of z^(1/3), z^(1/2) & ln(z) in Complex Plane

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SUMMARY

The discussion focuses on demonstrating the non-uniqueness of the complex functions z^(1/3), z^(1/2), and ln(z) in the complex plane. Daniel explains that expressing z in polar form as re^{iθ} reveals that z^(1/3) can take multiple values due to the periodic nature of the exponential function, specifically e^{i(θ + 2π)}. The conversation also touches on the need for understanding Laurent series, residues, and poles for undergraduate studies in complex analysis.

PREREQUISITES
  • Understanding of complex numbers and their polar representation
  • Familiarity with the properties of exponential functions in the complex plane
  • Basic knowledge of complex functions and their branches
  • Introduction to Laurent series and residue theory
NEXT STEPS
  • Study the properties of complex logarithms and their branches
  • Learn about the derivation and applications of Laurent series
  • Explore the concept of residues and how to calculate them
  • Investigate the implications of poles in complex analysis
USEFUL FOR

Undergraduate students in mathematics, particularly those studying complex analysis, as well as educators seeking to clarify concepts related to complex functions and their properties.

DanielO_o
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How does one show that z^{1/3} is not unique in the complex plane?

[ Similarly for z^(1/2) and ln(Z) ]


Thanks,

Daniel
 
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Write z as re^{i\theta} in "polar form". Then z^{1/3}= r^{1/3}e^{i\theta/3}. Now e^{i(\theta+ 2\pi)}= e^{i\theta} but e^{i(\theta+ 2\pi)/3} is not the same as e^{i\theta/3}.
 
Thanks :)
 
Dear Mentors,

Could anyone include explanations about the Laurent series, & the Residues & Poles ? Everything for an undergraduate course ?

I didn't find anything about that on the forum. if there's a good one please tell me.

Thank You in Advance

----------------
Yours Truly
BOB Merhebi
Astrobob Group
www.astrobob.tk[/URL]
 
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