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\title{Math 3160. Complex Variables, Spring 2018. Midterm exam 1. Version 2}
\author{Your name .............................................................................}
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\begin{enumerate}
\item Expess the absolute value and argument of $w$ through $x, y$ where $z= x+ i y$,
$$
w= \frac{1- \bar{z}}{1+ z}
$$

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\item Find all roots of the equation
$$ z^4= -16 $$

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\item Plot the region in the $(x, ~y)$ plane, where $x=\Re\, z$ and $y=\Im\, z $ are the real and imaginary parts of $z$:
$$ \Re \, z\geq | \Im \, z |, ~~~|z| < 2
$$

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\item Show that f(z) is analytic, and $\Im (f(z))$ is harmonic, compute the derivative $f'(z)$. 
$$
f(z)=\cos(z)= \cos(x) \cosh(y) - i \sin(x) \sinh(y). 
$$
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\item Compute 
$$ \log z, ~~ {\rm Log} (z), ~~\mbox{where }  z= (1-i)^4 $$

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\item Compute $$(1-i)^{i}$$

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\item Solve (find all solutions)
$$ (a) ~ \sin(z) = 5, \quad (b) ~ \exp(z) =-1
$$

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\item 
Express $\tanh (i z)$ through $\sin (z)$ and $\cos(z)$
\end{enumerate}



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