Hi Nikos. Thank you for your solution.
I would like to make a geometrical comment.
My opinion is that \(\displaystyle{k\in\mathbb{Z}}\) is the rotation index of the circle \(\displaystyle{S^1}\)
having the paramatrization \(\displaystyle{S^1=\left\{\left(\cos\,x,\sin\,x\right)\in\mathbb{R}^2: x\in\mathbb{R}\right\}}\).
For example, if we want to paramatrize the circle from \(\displaystyle{\left(\cos\,\phi,\sin\,\phi\right)}\) (\(\displaystyle{k=0}\)
then, \(\displaystyle{X\simeq Y=\left(\phi,\phi+2\,\pi\right)}\) by defining \(\displaystyle{f(x)=(\cos\,x,\sin\,x)\,,x\in Y}\) .
If we want to "run" the circle one time (\(\displaystyle{k=1}\)), then the above function with domain
\(\displaystyle{\left(\phi+2\,\pi,\phi+4\,\pi\right)}\), is appropriate.
What's your opinion ?
