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oab729
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Homework Statement
(z+1)^100=(z-1)^100 z is complex
Homework Equations
The Attempt at a Solution
(z-1)/(z+1)=e^(i2pi(k/100))
Is the Real Part of (z+1)^100 and (z-1)^100 Always Zero for Complex z?
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SUMMARY
The discussion centers on the equation \((z+1)^{100} = (z-1)^{100}\) for complex \(z\). The transformation \(\frac{(z-1)}{(z+1)} = e^{i2\pi(k/100)}\) is established, indicating that the ratio of the two expressions results in a complex exponential. By applying Euler's formula \(e^{ix} = \cos x + i \sin x\) and multiplying by the complex conjugate of the denominator, the real part of the expressions can be analyzed. The conclusion drawn is that the real part is not always zero, depending on the value of \(z\).
PREREQUISITES- Understanding of complex numbers and their properties
- Familiarity with Euler's formula \(e^{ix} = \cos x + i \sin x\)
- Knowledge of complex conjugates and their multiplication
- Basic algebraic manipulation of complex expressions
- Study the implications of complex exponentials in trigonometric identities
- Explore the geometric interpretation of complex numbers on the Argand plane
- Learn about the roots of unity and their applications in complex analysis
- Investigate the behavior of polynomial functions in the complex plane
Mathematics students, particularly those studying complex analysis, educators teaching advanced algebra, and anyone interested in the properties of complex functions.
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