6720-2022-08

**(a)** (8 points) Let \(P(z)\) be a polynomial of degree \(n\), with \(n\) simple roots, none of which lie on a simple closed contour \(C\). Evaluate \[ I=\frac{1}{2\pi i}\oint_C \frac{P'(z)}{P(z)}\,dz. \] **(b)** (8 points) Use Liouville's Theorem to prove the Fundamental Theorem of Algebra. Let \(P(z)\) be any polynomial of integer degree \(m\ge 1\). Then there is at least one point \(z=\alpha\) such that \(P(\alpha)=0\); that is \(P(z)\) has at least one root. **(c)** (9 points) Let \(f(z)\) be a meromorphic function, i.e. a function that only has poles in the finite plane, defined inside and on a simple closed contour \(C\), with no zeros or poles on \(C\). Let \(N\) and \(P\) be the number of zeroes and poles, respectively, of \(f(z)\) inside \(C\); where a multiple zero or pole is counted according to its multiplicity. Show that \[ I=\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=N-P. \] **(a)**

Recall the binomial expansion \[ (a+b)^n=\sum_{k=0}^n \binom{n}{k}a^{n-k}b^k, \] where \[ \binom{n}{k}=\frac{n!}{k!(n-k)!}. \] **(a)** (12 points) Use the binomial expansion and Cauchy's Integral Theorem to evaluate \[ \oint_C \left(z+\frac{1}{z}\right)^{2n}\frac{dz}{z} \] where \(C\) is the unit circle centered at the origin. **(b)** (13 points) Use this result to establish the following real integral formula: \[ \frac{1}{2\pi}\int_0^{2\pi}(\cos\theta)^{2n}\,d\theta =\frac{(2n)!}{4^n(n!)^2}. \] **(a)**

Use residue calculus to show that \[ \int_{-\infty}^{\infty}\frac{x\sin(ax)}{x^4+4}\,dx =\frac{\pi}{2}e^{-a}\sin(a),\qquad a>0. \] **(a)**

Use the keyhole contour below to show that on the principal branch of \(x^k\) \[ I(a)=\int_0^\infty \frac{x^{k-1}}{x+a}\,dx =\frac{\pi}{\sin(k\pi)}a^{k-1},\qquad 00. \] *Figure description: A keyhole contour about the positive real axis with outer arc \(C_R\), inner arc \(C_\epsilon\) about \(z=0\), a branch cut along the positive real axis ending at \(x=R\), and marked points \(z=-b\) and \(z=-a\) on the negative real axis.* **(a)**

Consider a source at \(z=-a\) and a sink at \(z=a\) of equal strength \(k\). **(a)** (12 points) Show that the associated complex potential is \[ \Omega(z)=k\log\left[\frac{z+a}{z-a}\right]. \] **(b)** (13 points) Show that the flow speed is \[ \frac{2ka}{\sqrt{a^4-2a^2r^2\cos(2\theta)+r^4}}, \] where \(z=re^{i\theta}\).