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\textbf{MAT334  Test 3, {March 27, 2018 }}\\
Last name  \dotfill  \\
First name \dotfill  \\
ID \dotfill   \\
Section \dotfill \\
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\# & Mark\\
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\textbf{1} [3 points] & \\
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\textbf{2} [3 points] & \\
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\textbf{3} [4 points] &\\
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\textbf{4} [5 points] & \\
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\textbf{5} [5 points] & \\
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\textbf{Total } [20 ] & \\
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\noindent
\textbf{1}.  Find the coefficient of $z^{-5}$ in Laurent expansion
of $z^3\cos(\frac{1}{z})$ at $z_0=0\ $ ?
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\noindent
\textbf{2}. Find the radius of convergence of the Taylor expansion
at $i$ of 
$$ f(z)=\frac{1}{\cos(z) - 0.5 }\ . $$
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\noindent
\textbf{$3^a$}. 
Find an explicit function expression of the power series $\sum_{n\ge2} n(n-1)z^{n+1}$
valid for $|z| < 1\ $. What is the region of analyticity of that function ?
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\noindent
\textbf{$3^b$}.
Evaluate $\sum_{n\ge2} \frac{n(n-1)3^{n+1}}{4^{n+1}}\ $ .
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\textbf{4}. Find $ \int _{\Gamma} \frac{e^{z^2}+z^2+1}{(z-\pi)^3}dz\ $, 
where $\Gamma$ is as in the picture below.
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$\includegraphics[ width=7cm, height=4cm]{final_curve.jpg} $
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%\textbf{5}. Evaluate $\int_\Gamma \frac{z}{e^{iz}-1}dz\ $, where $\Gamma$
%is as in the picture below.
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%$\includegraphics[ width=7cm, height=4cm]{final_curve.jpg} $
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\textbf{5}. Find $\int_0^{2\pi} \frac{d\theta}{5+4\sin(\theta)}\ $.
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