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 QUESTION

\begin{description}

\item[(a)]
 Evaluate the integral

 $\int_\gamma z^n\,dz$

 where n is any integer and $\gamma$ is the circle of radius r
 about the origin. (Note that you will have to treat the two cases
 $n=-1$ and $n \neq 1$ separately.)

\item[(b)]
 What does this tell you about the residue theorem?

\end{description}

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ANSWER

 \begin{description}

  \item[(a)]
  $z(t)=r e^{it},\ \ 0 \leq t \leq 2 \pi,\ \
  \frac{dz}{dt}=ire^{it}$

  \begin{eqnarray*}
  \int_\gamma z^n\,dz&=&\int_0^{2\pi}(re^{it})^r ire^{it}\,dt\\
  &=&ir^{n+1}\int_0^{2\pi}e^{i(n+1)t}\,dt\\
  &=&\left\{ \begin{array}{ll}
  ir^{n+1}\left[\frac{e^{i(n+1)t}}{i(n+1)}\right]_0^{2 \pi}=0 &\textrm{ for }n
  \neq -1\\
  i\left[t\right]_0^{2 \pi}=2 \pi i &\textrm{ for } n=-1
  \end{array}\right.
  \end{eqnarray*}

  \item[(b)]
  If we write $f(z)=\sum_{n=-\infty}^\infty c_{-n}z^n$ for $|z|
  \leq r$

  then $\int _\gamma f(z)\, dz=\sum_{n=-\infty}^{\infty}
  C_{-n}\int_\gamma z^n\,dz =2 \pi i c_{-1}$

  \end{description}


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