Solving Complex Function for multiple solutions

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SUMMARY

The discussion focuses on solving the equation z4 + j4 = 0, which can be simplified to z4 = -1. Participants suggest using polar coordinates to express z, leading to the formulation zn = |a|ej(Θ + 2πk). The key takeaway is that the solutions can be derived by finding the fourth roots of -1, which involves determining the magnitude and angle in the complex plane.

PREREQUISITES
  • Understanding of complex numbers and polar coordinates
  • Familiarity with Euler's formula ejθ = cos(θ) + j sin(θ)
  • Knowledge of roots of unity in complex analysis
  • Basic skills in solving polynomial equations
NEXT STEPS
  • Study the concept of roots of unity in complex numbers
  • Learn how to convert complex numbers to polar form
  • Explore the application of Euler's formula in solving equations
  • Investigate the geometric interpretation of complex solutions in the Argand plane
USEFUL FOR

Mathematicians, engineering students, and anyone interested in complex analysis and solving polynomial equations in the complex plane.

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1. Find all the solutions to the equation z^4 + j^4 = 0



2. z^n = |a|e^j(Θ + 2pik)



3. I really don't know where to start, I thought about j^4 = 1, so z^4=-1. I then simplified to conclude that z^4 = -e^jpi. I am not sure if that is correct and if it is what to do next.
 
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You can write z in polar coordinates, too, and solve for r and possible values of Θ.
 

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