# Circular contour integration

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.

## 1 Answer

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?

• hey thanks but what do the question marks in the answer mean? Nov 18, 2017 at 17:12
• I gave you a hint. Are you able to complete the solution? Nov 18, 2017 at 17:14
• 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. Nov 18, 2017 at 17:23
• On the right side, you have to determine the coefficient of $1/z$ of the given function. Nov 18, 2017 at 17:27
• I think I got it thank you would it be A if n=1 B if n=2 C if n=3 Nov 18, 2017 at 17:43