MAT334-2018F > Final Exam

FE-P2

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Victor Ivrii:
(a)--(b) If $|z|=r$ and $\arg{w}=t$, then $z=r e^{it}$ and
\begin{align*}
w= \frac{1}{2}\bigl(re^{it} +r^{-1}e^{-it}\bigr)&= \cosh(s)\cos(t)+i\sinh(s)\sin (t)
\end{align*}
with $s=\ln (r)$ (and $r=e^s$).

Then $w=u+iv$ with $\frac{u^2}{a^2}+\frac{v^2}{b^2}=1$, $a=\cos(s)$, $b=\sinh(s)$ and also
$\frac{u^2}{A^2}-\frac{v^2}{B^2}=1$,
with  $A=\cos (\theta)$, $B=\sin(\theta)$, $A^2+B^2=1$.

(c) Consider, what the circle $\{z\colon |z|=1\}$ is mapped to: $z=e^{it}$, then
$w =\frac{1}{2}(e^{it}+e^{-it})=\cos(t)$ and it runs a line segment $[-1,1]\subset\mathbb{R}$. So $f$ maps the unit disk onto compliment of it $\mathbb{C}^*\setminus [-1,1]$ (and $0$ is mapped to $\infty$).

(d) See attachment


(e) $w=\frac{1}{2}(z+z^{-1}) \implies z^2 -2wz +1=0$; it has two roots $z_1$ and $z_2$ s.t.
$z_1z_2=1$ and $z_1+z_2=2w$; exactly one of them for $w\notin [-1,1]$ is in unit disk, and the second one is in $\{z\colon  |z|>1\}$.

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