6720-2024-08

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 \[ \bigl ||z_1|-|z_2|\bigr |\le |z_1+z_2|\le |z_1|+|z_2|. \]

(20 pts) Prove the Cauchy Integral Formula: Let \(f\) be analytic in an open domain \(D\), and let \(z\in D\). Then for any non-negative integer \(n\), \[ f^{(n)}(z)=\frac {n!}{2\pi i}\oint _C \frac {f(w)}{(w-z)^{n+1}}\,dw, \] where \(C\) is any simple contour in \(D\) enclosing \(z\). You may take the \(n=0,1\) versions of this formula as given.

Complete the following:

1. (10 pts) Prove Rouche’s theorem: Assume \(f\) and \(g\) are analytic functions inside and on a simple closed contour \(C\), and suppose \(|f(z)|>|g(z)|\) for all \(z\in C\). Then \(f\) and \(f+g\) have the same number of zeros inside \(C\).
2. (10 pts) Prove the following variant of the Argument Principle: Let \(C\) be a simple closed curve with counterclockwise orientation, and suppose \(f\) is analytic inside and on \(C\), except at \(M\) poles inside \(C\) (poles are counted with weight corresponding to their multiplicity). Let \(a\in \mathbb {C}\), and assume \(f(z)+a\) has no zeros on \(C\) and has \(N_a\) zeros inside \(C\) (again, counting according to multiplicity). Without citing the Argument Principle, prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)+a}\,dz=N_a-M. \]

1. (10 pts) Show that \[ \int _{-\infty }^{\infty }\frac {\cos x}{x^2-\frac {\pi ^2}{4}}\,dx=-2. \]
2. (10 pts) Compute the complete infinite asymptotic expansion of \[ I(k)=\int _0^\pi e^{-kt^2}t^{-3/4}\sin t\,dt. \]