Complex Numbers and Elementary Functions

Complete the following:

1. (10 pts) Use de Moivre’s Theorem to show that the two expressions, \[ \cos (n\theta ),\qquad \frac {\sin ((n+1)\theta )}{\sin \theta }, \] for arbitrary \(n\in \mathbb {N}\), are both degree-\(n\) polynomials of \(\cos \theta \).
2. (10 pts) Prove the triangle inequality: Given \(z_1,z_2\in \mathbb {C}\), then \[ \bigl ||z_1|-|z_2|\bigr |\le |z_1+z_2|\le |z_1|+|z_2|. \]

Let \(X\) be an inner product space with inner product \(\langle \cdot ,\cdot \rangle \). Let \((a_j)\) be a sequence of complex numbers and let \((\psi _j)\) and \((\phi _j)\) be two orthonormal sequences in \(X\). Let \(T:X\to X\) be an operator defined by \[ Tx=\sum _{j=1}^{\infty } a_j\langle \psi _j,x\rangle \phi _j. \] State and prove a (nontrivial) sufficient condition on \((a_j)\) so that \(T\) is a compact operator.

Suppose \(f(z)\) is analytic in some neighborhood of the point \(z_0\) without the point \(z_0\) itself; let \(z_0\) be an essential singularity of \(f(z)\).

Show that for any complex number \(C\), there is a sequence of points \(z_n\) (\(n=0,1,2,\ldots \)) such that \[ z_n\to z_0\qquad \text {and}\qquad f(z_n)\to C. \] (In other words, in every neighborhood of an essential singularity, the function \(f(z)\) is arbitrary close to every complex number.)