Why is this particulary true? There is no exaplanation in my notes, to one of the solutions and just says:
By Cauchy's Theorem,for $R> r$ $$\oint_{C_r}\frac{dz}{P(z)}=\oint_{C_R}\frac{dz}{P(z)} $$
I do not understand how this holds, can someone please explain why this is the case?
EDIT: $P(z)$ is a polynomial of degree $n\ge 2$.
Well, there are lots of details missing, but my best guess is that the loops $\gamma_r\colon[0,2\pi]\longrightarrow\mathbb C$ and $\gamma_R\colon[0,2\pi]\longrightarrow\mathbb C$ defined by $\gamma_r(t)=re^{it}$ and $\gamma_R(t)=Re^{it}$ respectively are homotopic in $\mathbb{C}\setminus\{\text{roots of }P\}$. That is, $P$ has no roots whose absolute valu belongs to $[r,R]$.