How to compute a certain contour integral.

Question

How can I compute the contour integral $ \displaystyle \oint_{C} \frac{1}{z^{3} + 9 z} \, d{z} $, where $ C := \{ z \in \mathbb{C} \,|\, |z| = 4 \} $ is the counterclockwise-oriented circle with center at $ 0 $ and with radius $ 4 $?

Which method of integration should I choose? Any help would be deeply appreciated.

Answer 1

Use the Residue theorem, which as you will recall states that the value of a contour integral around a closed path is $i 2 \pi$ times the sum of the residues of the poles of the integrand of the contour integral that are contained within the closed path.

So you need to know the poles, which are the zeroes of the denominator of the integrand. What are these? And once you determine these, which ones, if any, are located within the given closed path, i.e., $|z|=4$.

Once you have the poles inside that closed path, you need to compute the residues. In this case, the residue of a function $f(z)=1/q(z)$ at a pole $z=z_0$ is $1/q'(z_0)$. Compute this for each pole inside the closed path. Sum the residues. Multiply by $i 2 \pi$. Done.

Answer 2

HINT

$$ \int_C\frac{1}{z^3+9z}dz=\int_C\frac{1}{z(z^2+9)}dz=\int_C\frac{1}{z(z+3i)(z-3i)}dz. $$

Let $f(z)=\frac{1}{z(z+3i)}$, $g(z)=\frac{1}{(z+3i)(z-3i)}$, $h(z)=\frac{1}{z(z-3i)}$. Using Cauchy's Integral Formula, we can use $f(z)$, $g(z)$, and $h(z)$ to get the final result.