# Toronto Math Forum

## MAT334-2018F => MAT334--Tests => Quiz-1 => Topic started by: Victor Ivrii on September 28, 2018, 04:17:06 PM

Title: Q1: TUT 5101
Post by: Victor Ivrii on September 28, 2018, 04:17:06 PM
$\renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}}$
Find all solutions of the given equation.
\begin{equation*}
(z+1)^2 =1-i.
\end{equation*}
Title: Re: Q1: TUT 5101
Post by: Mengyang Li on September 28, 2018, 08:16:23 PM
Solution is scanned
Title: Re: Q1: TUT 5101
Post by: Victor Ivrii on September 29, 2018, 03:38:51 PM
Still, typed is better
Title: Re: Q1: TUT 5101
Post by: Yatong Yu on September 29, 2018, 05:58:01 PM
$(z+1)^2=1-i$ which is $(1, -1)$ on the axis with angle $-\pi /4$ and length $2^{1/2}$ so $1-i = 2^{1/2}e^{i(-\pi /4+2k\pi )}$
$$z+1=(1-i)^{1/2} =\bigl(2^{1/2}e^{i(-\pi /4+2k\pi )}\bigr)^{1/2}$$ $$z=2^{1/4}e^{i(-\pi /8 + k\pi )}-1$$
when $k=0$   $z=2^{1/4}e^{-i\pi /8}-1=2^{1/4}(\cos \pi /8 - i\sin \pi /8)-1$
when $k=1$   $z=2^{1/4}e^{i7\pi /8}-1=-2^{1/4}(\cos \pi /8 - i\sin \pi /8)-1$

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