Null-homotopic function on a set

Question

If we consider the path $γ : [0,1] → \mathbb{C}$ with $t → −i + 2e^{2πit}$. Is $γ$ null-homotopic on the following set: $Ω = \{ z∈\mathbb{C} : 1/2 <|z|< 4 \}$?

I have that it is, but I am not sure - could someone tell me if I am correct?

Answer 1

No, it is not. If it was, then the integral of any analytic map from $\Omega$ into $\Bbb C$ along $\gamma$ woud be zero. But$$\int_\gamma\frac{\mathrm dz}z=2\pi i,$$since$$\frac1{2\pi i}\int_\gamma\frac{\mathrm dz}z$$is the winding number of $\gamma$ with respect to $0$, which is equal to $1$.