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\title{ Midterm 2}
\author{M 3160: Applied Complex Variable }
\date{Spring 2018}
\maketitle

\begin{enumerate}
\item 
Evaluate the integral
$$
I_1=\int_C \frac{\exp(z)}{(z+8)(z-1)} dz, \quad C:=\{ z: |z| =2 \}.
$$ 
\newpage
\item 
Evaluate the integral
$$
I_2=\int_C \frac{z^2-\exp(2z)}{(z+i)^3} dz, \quad C:=\{ z: |z| =2 \}
$$ 
\newpage
\item 
Expand  function 
$$ F(z)= \frac{1}{2+z} $$ into the Taylor series around the point $z_0=1$. What is the radius of convergence? 
\newpage
  \item
  Expand function
  $$
  f(z)= \cos(2z)- \cos\left( \frac{2}{z} \right)
  $$
   into the Laurent series around the point $z_0=0$. 
   \newline
   Find the coefficients of the expansion. 
      \newline
      What is the radius of convergence? 
\newpage
\item 
Using representation
$$
F(z)=\frac{3 z}{(z-1)(z+2)}= \frac{1}{z-1}+ \frac{2}{z+2},
$$
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.
\newpage
\item 
Compute the residue 
$$ {\rm Res}_{z=0} f(z) $$ if the function $f(z)$ is either
$$f(z)= f_a(z)= \frac{e^{i z}-z^3}{z^4} ~(a); \quad \mbox{or }f(z)=f_b(z)= \cos\left( \frac{1}{z} \right) ~ (b) . $$


\newpage
\item 
Indicate the type of singularity of the following functions at $z=z_0$. Is is
\begin{enumerate}
\item a removable singular point,
\item  a pole of the order $n$ (find $n$), or
\item An essential singular point?
\end{enumerate}
Consider the functions
\begin{eqnarray*}
&(1) ~f_1(z)= \frac{z- \sin(z)}{z^3}, ~z_0=0; \quad & (2) ~f_2(z)= z^5 \sinh\left(\frac{1}{z^2} \right),~z_0=0;
\\
&(3) ~f_3(z)= \frac{1- \cos(z)}{z^3}, ~z_0=0; \quad  &(4) ~f_4(z)= \frac{\exp(iz)}{(z-2)^2}, ~z_0=3.
\end{eqnarray*} 

\end{enumerate}


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