Author Topic: TT1 Problem 5 (night)  (Read 774 times)

Victor Ivrii

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TT1 Problem 5 (night)
« on: October 19, 2018, 04:18:00 AM »
$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
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

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Re: TT1 Problem 5 (night)
« Reply #1 on: October 19, 2018, 09:32:23 AM »
w= ez
∴w= e(x+yi)=ex∙eyi
    =ex(cosy + isiny)
∴w = excosy + iexsiny
∴(excosy)2 +(iexsiny)2≥ 22
∴e2x(sin2y+cos2y)≥ 4
∴e2x≥4
∴x≥ln4/2=ln2
also excosy ≥ 0 => cos y ≥ 0 => π/2≥y≥0
exsiny≥ 0 =>siny ≥0 =>π ≥y ≥ 0
∴ {Z: Z = x + yi, x≥ ln2, π/2≥y≥0}

Victor Ivrii

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Re: TT1 Problem 5 (night)
« Reply #2 on: October 20, 2018, 02:54:42 PM »
It is way better to use MathJax/LaTeX than ugly, non-portable and much more limited facilities of html