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Let $\gamma$ be the circular contour, positively oriented, with centre $0$ and radius $7$. Let $A$, $B$ and $C$ be complex numbers. Compute the following integral $$\int_{\gamma } \frac{A+Bz+Cz^2}{z^n} dz.$$

I have no clue where to start with this question if anybody could help me.

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1 Answer 1

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Hint. Recall that the residue at $0$ of a function $f$ is the coefficient of $1/z$ in the Laurent series expansion of $f$ centered at $0$. In this case, the given function is already expanded and therefore, by the Residue Theorem, $$\frac{1}{2\pi i}\int_{|z|=7 } \frac{A+Bz+Cz^2}{z^n} dz=\begin{cases} A&\text{if $n=1$,}\\ ?&\text{if $n=2$,}\\ ?&\text{if $n=3$,}\\ 0&\text{otherwise.}\\ \end{cases}$$ Are you able to complete the evaluation?

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  • $\begingroup$ hey thanks but what do the question marks in the answer mean? $\endgroup$
    – RodgerS
    Nov 18, 2017 at 17:12
  • $\begingroup$ I gave you a hint. Are you able to complete the solution? $\endgroup$
    – Robert Z
    Nov 18, 2017 at 17:14
  • $\begingroup$ I dont think I would be able to Its just a bit confusing for me, I'm better at seeing the final solution and working backwards to understand something :/ I'm sorry. $\endgroup$
    – RodgerS
    Nov 18, 2017 at 17:23
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    $\begingroup$ On the right side, you have to determine the coefficient of $1/z$ of the given function. $\endgroup$
    – Robert Z
    Nov 18, 2017 at 17:27
  • $\begingroup$ I think I got it thank you would it be A if n=1 B if n=2 C if n=3 $\endgroup$
    – RodgerS
    Nov 18, 2017 at 17:43

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