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

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Discussion Overview

The discussion centers on the uniqueness of the complex functions z^{1/3}, z^{1/2}, and ln(z) within the complex plane. Participants explore the implications of these functions' multi-valued nature and their representation in polar form.

Discussion Character

  • Exploratory, Technical explanation

Main Points Raised

  • One participant questions how to demonstrate that z^{1/3} is not unique in the complex plane.
  • Another participant suggests using polar form, indicating that writing z as re^{iθ} leads to the conclusion that z^{1/3} can take multiple values due to the periodicity of the exponential function.
  • The same participant notes that while e^{i(θ + 2π)} equals e^{iθ}, the expression e^{i(θ + 2π)/3} does not equal e^{iθ/3}, highlighting the non-uniqueness.
  • A separate post by another participant requests information about Laurent series, residues, and poles, indicating a desire for foundational knowledge relevant to undergraduate studies.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the uniqueness of the functions discussed, as the conversation includes both a specific inquiry about z^{1/3} and a separate request for information on related topics.

Contextual Notes

The discussion does not address the assumptions underlying the uniqueness of these functions or the implications of their multi-valued nature in detail.

Who May Find This Useful

Individuals interested in complex analysis, particularly those studying the properties of multi-valued functions and seeking foundational knowledge in related concepts such as Laurent series and residues.

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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