I am having trouble trying to solve the following contour integral for an assignment: $\oint_C \left(z + \frac{1}{z^2}\right) \, dz$ where $C$ is the contour defined by $|z| = 2$. My professor is known for assigning tricky problems, and I wonder if this is one of them, as we have been specifically asked to use Cauchy's Theorem to solve it.
I know that typically Cauchy's Theorem is applied to analytic functions without singularities inside the contour of integration. However, the function $f(z) = z + \frac{1}{z^2}$ clearly has singularities at $z = 0$.
I have reviewed my notes and understand that the Cauchy Integral Formula is a common way to solve such integrals when there are singularities. However, I would like to clarify that I am specifically exploring whether there is a way to use Cauchy's Theorem for this exercise.
I appreciate in advance any help or advice you can provide. If you need more details about the exercise or any further clarification, please let me know.
No, Cauchy's first integral theorem cannot be used directly here. It states that if $U$ is a simply connected domain, and $C$ is a closed curve in $U$ and $f$ is holomorphic on $U$, then $$\int_C fdz=0.$$ $C$ is obviously closed and $f$ is holomorphic on $C$ but we need to choose $U$. The problem is that we cannot choose $U$ to be simply connected without including $0$. However if we include $0$ then $f$ is not holomorphic on all of $U$.
In this case $f$ has a primitive on an open set containing the curve. In particular $$F(z)=\frac{z^2}2-\frac1z.$$ We can thus evaluate the integral as $$F(C(1))-F(C(0))=F(2)-F(2)=0$$ if we assume that $C$ starts and ends at the point $2$.