Solving Complex Variables Homework

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SUMMARY

This discussion focuses on solving two complex variable problems using De Moivre's theorem and the geometric series formula. The first problem requires demonstrating that the sum of all n values of z^(1/n) equals zero for n >= 2, while the second problem involves using these concepts to derive a specific formula. Participants emphasize the importance of manipulating equations correctly and equating terms to find solutions. The discussion highlights common challenges faced when applying these mathematical principles.

PREREQUISITES
  • Understanding of De Moivre's theorem
  • Familiarity with geometric series formula
  • Basic knowledge of complex numbers
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Study the proof of De Moivre's theorem in detail
  • Practice solving problems involving geometric series
  • Explore applications of complex numbers in different mathematical contexts
  • Learn techniques for manipulating algebraic equations effectively
USEFUL FOR

Students studying complex variables, mathematics educators, and anyone seeking to enhance their understanding of De Moivre's theorem and geometric series in complex analysis.

JasonPhysicist
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Homework Statement


I'd like some help with 2 problems:

Show by using Demoivre's theorem and the geometric series formula that the sum of all n values of z^(1/n) is zero when n >=2.
Z is a complex number.

Use the geometric series formula and Demoivre's theorem to show that:

eq3.png

Homework Equations



the geometric series formula:
eq1.png


Demoivre's theorem

eq2.png

The Attempt at a Solution



For the first part,I've tried to make z^(1/n) = p so that p^n = z ,but I had no success showing that the sum equals zero...
For the second part I've made z= cos(theta) + i sin(theta) and I've obtained the left part of the formula,but I can't get the right part...

I'd appreciate any help,because I don't seem to be going anywhere.
Thank you in advance!
 
Last edited:
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If z = cos(\theta), what is z2, z3, and so on?
 
For the second question, a big hint is to equate equivalent terms.

a + bi = c + di --> a = c, b = d

Don't move things across the equals sign, but work on each side separately
 

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