TT1 Problem 5 (night)

[Victor Ivrii]

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

[Yatong Yu]

w= e^z
∴w= e^(x+yi)=e^x∙e^yi
    =e^x(cosy + isiny)
∴w = e^xcosy + ie^xsiny
∴(e^xcosy)² +(ie^xsiny)²≥ 2²
∴e^2x(sin²y+cos²y)≥ 4
∴e^2x≥4
∴x≥ln4/2=ln2
also e^xcosy ≥ 0 => cos y ≥ 0 => π/2≥y≥0
e^xsiny≥ 0 =>siny ≥0 =>π ≥y ≥ 0
∴ {Z: Z = x + yi, x≥ ln2, π/2≥y≥0}

[Victor Ivrii]

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