Complex numbers and ODEs: problem set 1 (on de Moivre's theorem and the roots of unity)

4. Find the shape of complex functions

(vi) Find the shape of the graph of $|z + 1| + |z - 1| = 8$.

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(viii) Find the graph of $\textrm{arg}[(z-4)/(z-1)] = 3\pi/2$

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5. Application of de Moivre's theorem

Use de Moivre's theorem to prove that \[ \cos 4\theta = 8\cos^4 \theta - 8 \cos^2 \theta + 1. \]

Deduce expressions for $\cos(\pi/8)$ and $\cos(3/\pi/8)$.

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9. Proof for the roots of unity

(i) Show that the sum of the $n$ roots of the equation $z^n = a \in \mathbb{C}$ is zero.

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(ii) By consider the roots of $z^{2n+1} + 1 = 0$, show that \[ \sum_{k=-n}^{n} \cos \left( \frac{2k+1}{2n+1}\right) = 0. \]

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10. Solving an equation

Find the roots of $(z-1)^n + (z + 1)^n = 0$.

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12. Binomial Theorem proof

Prove that \[ \sum_{r=1}^n \binom{n}{r} \sin (2r\theta) = 2^n \cos^n \theta \sin(n\theta) \]

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