Argue informally that $\int_{γ}\frac{1}{z^2}dz=0$

Question

Let $γ$ be a simple closed curve contenting $0$ Argue informally that $\int_{γ}\frac{1}{z^2}dz=0$

I think this is simple if the integral is equal to $0$ then the antiderivative $-\frac{1}{z}$ would be holomorphic on $0$ and that's a contradiction.

My question is if I couldn't find the antiderivative how could I justify that the integral wouldn't be $0$?

Answer 1

You wrote that “if the integral is equal to $0$ then the antiderivative $−\frac1z$ would be holomorphic on $0$ and that's a contradiction”. There are lots of problems here. The function $z\mapsto-\frac1z$ is holomorphic on $\Bbb C\setminus\{0\}$; there is no “if” here. And, since $z\mapsto\frac1{z^2}$ has an antiderivative on $\Bbb C\setminus\{0\}$, its integral along any closed path is $0$.

And if you can't find an antiderivative of a function $f$, then it still may well be true that $\int_\gamma f=0$ for any closed path $\gamma$.