Toronto Math Forum
MAT3342018F => MAT334Tests => Term Test 1 => Topic started by: Victor Ivrii on October 19, 2018, 04:18:00 AM

$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Find any region that is mapped bijectively (onetoone) to $\{w\colon \Re w\ge 0,\ \Im w\ge 0, \ w\ge 2\}$ by the map $w=e^z$. Draw both of them.

w= e^{z}
∴w= e^{(x+yi)}=e^{x}∙e^{yi}
=e^{x}(cosy + isiny)
∴w = e^{x}cosy + ie^{x}siny
∴(e^{x}cosy)^{2} +(ie^{x}siny)^{2}≥ 2^{2}
∴e^{2x}(sin^{2}y+cos^{2}y)≥ 4
∴e^{2x}≥4
∴x≥ln4/2=ln2
also e^{x}cosy ≥ 0 => cos y ≥ 0 => π/2≥y≥0
e^{x}siny≥ 0 =>siny ≥0 =>π ≥y ≥ 0
∴ {Z: Z = x + yi, x≥ ln2, π/2≥y≥0}

It is way better to use MathJax/LaTeX than ugly, nonportable and much more limited facilities of html