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{\bf Math 122-A \hskip.1in   Winter 2022\hskip.1in HOMEWORK\# 4 (due February. 10)
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\begin{enumerate}
    \item In each case find all the values of $z$ such that:
    
    \begin{enumerate}
        \item  $z=i^{i}$
\item  $z=(1-i)^{1+i}$
\item  $e^{1 / z}=1+i \sqrt{3}$.
        
    \end{enumerate}

    \item For any $z \in \mathbb{C}$ define:
$$
\sin (z)=\frac{e^{i z}-e^{-i z}}{2 i}, \cos (z)=\frac{e^{i z}+e^{-i z}}{2}, \sinh (z)=\frac{e^{z}-e^{-z}}{2}, \cosh (z)=\frac{e^{z}+e^{-z}}{2}
$$
Prove the following identities hold. 
\begin{enumerate}


    
\item $\sin (-z)=-\sin (z), \quad \cos (-z)=\cos (z)$,
\item  $\sin ^{2}(z)+\cos ^{2}(z)=1, \quad \cosh ^{2}(z)-\sinh ^{2}(z)=1$,
\item $\cos \left(z_{1}+z_{2}\right)=\cos \left(z_{1}\right) \cos \left(z_{2}\right)-\sin \left(z_{1}\right) \sin \left(z_{2}\right)$,
\item $\cosh \left(z_{1}+z_{2}\right)=\cosh \left(z_{1}\right) \cosh \left(z_{2}\right)+\sinh \left(z_{1}\right) \sinh \left(z_{2}\right)$,
\item $(\sin (z))^{\prime}=\cos (z), \quad(\cos (z))^{\prime}=\sin (z)$,
\item $(\sinh (z))^{\prime}=\cosh (z), \quad(\cosh (z))^{\prime}=\sinh (z)$,
    
    
    
\end{enumerate}

    \item 

 Evaluate the following integrals:
\begin{enumerate} 

\item $$\int_{1}^{2}\left(\frac{1}{t}+i\right)^{2} d t,$$ 
\item $$\int_{0}^{\pi / 3} e^{i t} d t,$$
\item $$\int_{0}^{2 \pi} e^{i k t} d t, \quad k \in \mathbb{Z}.$$

\end{enumerate}

    \item  In each case write the equation of the curve representing:
    
    \begin{enumerate}
        \item 
  
 the segment joining $i$ and $i$,
\item the circumference of center $1-i$ and radius 2, in the counter clockwise,
\item the triangle with vertices $1, i,-2$
      \end{enumerate}


\item  Evaluate the following integrals:

\begin{enumerate}
    

\item  $\int_{\gamma} x d z, \quad \gamma$ the boundary on the unit square
\item  $\int_{\gamma} e^{z} d z, \quad \gamma$ the portion of the unit circle joining 1 and $i$ in the counter clockwise direction.
\item  $\int_{\gamma} x d y, \gamma$ is the boundary of a bounded region $A \subset \mathbb{R}^{2}$ (without holes) in the counter
clockwise direction. HINT : use Green's Theorem.
\end{enumerate}
\end{enumerate}




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