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Toronto Math Forum
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MAT334-2018F
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MAT334--Tests
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Term Test 1
»
TT1 Problem 5 (night)
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Topic: TT1 Problem 5 (night) (Read 5967 times)
Victor Ivrii
Administrator
Elder Member
Posts: 2607
Karma: 0
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.
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Yatong Yu
Jr. Member
Posts: 6
Karma: 6
Re: TT1 Problem 5 (night)
«
Reply #1 on:
October 19, 2018, 09:32:23 AM »
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}
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Victor Ivrii
Administrator
Elder Member
Posts: 2607
Karma: 0
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
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Toronto Math Forum
»
MAT334-2018F
»
MAT334--Tests
»
Term Test 1
»
TT1 Problem 5 (night)