Finding Branch Points & Cuts for 3 Functions

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SUMMARY

The discussion focuses on identifying branch points and introducing branch cuts for three specific complex functions: a) \( \frac{1}{4+\sqrt{z^2-9}} \), b) \( \sqrt{4+\sqrt{z^2-9}} \), and c) \( \ln[5+\sqrt{\frac{z+1}{z-1}}] \). Participants express confusion regarding the methodology for determining branch points, particularly in relation to the function \( f(z)=\sqrt{z} \) which branches at 0. The need for a clear, systematic approach to identifying these points is emphasized, as well as the importance of understanding the underlying principles of complex analysis.

PREREQUISITES
  • Complex analysis fundamentals
  • Understanding of branch points and branch cuts
  • Knowledge of square root and logarithmic functions in the complex plane
  • Familiarity with evaluating limits and singularities
NEXT STEPS
  • Study the concept of branch points in complex functions
  • Learn how to determine branch cuts for functions like \( \sqrt{z} \) and \( \ln(z) \)
  • Explore the implications of branch cuts on function continuity and evaluation
  • Review examples of complex functions with known branch points and cuts
USEFUL FOR

Students of complex analysis, mathematicians, and anyone seeking to deepen their understanding of branch points and cuts in complex functions.

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Homework Statement


Find the branch points and introduce branch cuts for the below functions:

a) [tex]\frac{1}{4+\sqrt{z^2-9}}[/tex]

b) [tex]\sqrt{4+\sqrt{z^2-9}[/tex]

c) [tex]ln[5+\sqrt{\frac{z+1}{z-1}}][/tex]

Homework Equations


The Attempt at a Solution



So, the professor explained branches and branch cuts like they were completely obvious to him (which, they probably are...to him). He didn't show us basically at all how to find branching points and branch cuts except by giving us an example and finding it "obviously". Like he said the function f(z)=sqrt(z) branches at 0...but he never told us how we should find that point. I have no idea how to show where the branch points are...are they points where the function evaluates to 0 or something? Someone help?
 

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