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\title{M 3160 Applied Complex Variable: Spring 2018\\ Final exam}
\date{}
\author{Your name ...................................................}

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Solve any six problems. 
\begin{enumerate}
\item 
Find all roots of the equation, plot then on the complex plane.
$$ z^3+ 8 i=0 $$

\newpage 
\item The real part of a complex function $f(x, y)=u(x, y)+i v(x, y)$ is
$$
u(x, y)=x^3- 3 x y^2
$$

a) Check that $u(x, y)$ is a harmonic function.

b) If  $f(x,y)$ is an analytic function, what is its imaginary part $v(x, y)$?
\newpage
\item 
Solve the equation; find all solutions.
$$ e^z+1=0 $$

\newpage
\item 
Using Cauchy integral formula, evaluate the integrals
$$
I_1=\int_C \frac{\exp(z^2-10)}{(z+8)(z-1)} dz, \quad C:=\{ z: |z| =3 \}.
$$ 
$$
I_2=\int_C \frac{\exp(z^2-10)}{(z+8)(z-4)} dz, \quad C:=\{ z: |z| =3 \}.
$$

\newpage
\item 
Using representation
$$
F(z)=\frac{5}{(z-2)(z+3)}= \frac{1}{z-2}- \frac{1}{z+3}
$$
find the Laurent series of  function $F(z) $ around the point $z_0=0$. 
\newline Define three regions where three different expansions are valid, find these expansions.
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\newpage
\item 
Using residue theorem, evaluate integral
$$
I_2=\int_0^\infty \frac{dz}{z^4+16} dz,
$$ 
\newpage

 \item
 Find a linear fractional transformation that maps the horizontal line $[ -\infty< x< \infty, ~y=1]$ onto the circle $|u+iv|=1$. 



\end{enumerate}


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