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\begin{center}
{\huge  \sc  Homework 5 } \\
\smallskip
{ Date: 2/11/21}\\
\smallskip
{Name:  }\\
{\large  }
\end{center}

\begin{enumerate}
\item Compute the principle value of the arguments of the following complex numbers:\newline 
$ 1+i, \frac{1}{2}+i\frac{\sqrt{3}}{2}, (1+i)^3, (\frac{1}{2}+i\frac{\sqrt{3}}{2})^{243}, (1+i)^2(\frac{1}{2}+i\frac{\sqrt{3}}{2})^3 $


\item Let $\arg{z}$ denote the principle value of the argument for $z\neq 0$, that is, $-\pi < \arg{z} \leq \pi$. For real, $x,y$ with $x<0$ show:
\begin{enumerate}[(i)]
    \item $\lim_{y\rightarrow 0}\arg{(x+i|y|)}=\pi$
    
    \item $\lim_{y\rightarrow 0}\arg{(x-i|y|)}=-\pi$
    
    Compute the corresponding limits when $x=0$ and when $x>0$.
\end{enumerate}

\item Compute the following principal logarithms: $ Log{(3i)},Log(-2i), Log(1+i), Log(-1), Log(z^{10})$ 

where $z=2e^{i\pi/3}, Log(x)$ for real $x\neq 0$. 

\item For $z_1,z_2\neq 0$ show that
\[
Log(z_1z_2)=Log(z_1)+Log(z_2)+2n\pi i
\]
where $n$ is an integer that need not be zero. Specify the value that $n$ may take. Show that a logarithm of $z_1z_2$ is of the form
\[
\log(z_1)+ \log(z_2)
\]
provided that appropriate values of the logarithms are taken. 
\item Let $n$ be a positive integer. A complex number $\omega$ is said to be an nth roof of unity if \newline 
$\omega^n=1$.
\begin{enumerate}[(i)]
    \item Find all nth roots of unity in polar coordinates and draw a picture. 
    
    \item For $n=2,3,4$ express the nth roots of unity in the form $x+iy$. 
    
    \item If $\omega_1,\omega_2$ are nth roots of unity, show that the following are also nth roots of unity:
    \[
    \omega_1^n \qquad \omega_1\omega_2 \qquad \omega_1/\omega_2
    \]
    
    \item For given $r,\theta \in \mathbb{R}(r>0)$, find all $z\in \mathbb{C}$ such that 
    \[
    z^n=re^{i\theta}
    \]
    
    \item If $z_1^n=z_2^n$, show that $z_1=\omega z_2$ where $\omega$ is an nth root of unity.
\end{enumerate}
\item For $z, \beta\in \mathbb{C}, z\neq 0$, define the principal value of $z^{\beta}$ to be
\[
z^{\beta}=e^{\beta Log(z)}
\]
Compute the principal values of the following powers:
\[
1^{\sqrt{2}} \qquad (-2)^{\sqrt{2}}\qquad i^i \qquad 2^i \qquad (3-4i)^{1+i} \qquad (3+4i)^5
\]
\newpage
\item Draw the following paths and specify all the continuous choices of arguments along them:


\begin{enumerate}[(i)]

\item $\gamma(t)=2e^{-it} (t\in [0,4\pi])$

\item $\gamma(t)=t+i(1-t) (t\in[0,1]$

\item $\gamma(t)=t-1+it^2 (t\in [-1,1])$

\item $\gamma(t)=\begin{cases} t+i(1-t)  &\mbox{if } t\in[0,1] \\
1+i(1-t) & \mbox{if } t\in[1,2] \end{cases}$

In each case compute the winding number of the path round the origin. 
\end{enumerate}
   
    

\end{enumerate}

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