On the third edition of Ahlfors' Complex Analysis, page 149 it states: (i) Draw a circle $C_j$ about $a_j$ of radius $<\delta_j$, and let $P_j=\int_{C_j}f(z)dz$ be the corresponding period of $f(z)$. (ii) The particular function $1/(z-a_j)$ has the period $2\pi i$. (iii) Therefore, if we set $R_j=P_j/2\pi i$, the combination $f(z)-\frac{R_j}{z-a_j}$ has a vanishing period. The constant $R_j$ which produces this result is called the residue of $f(z)$ at the point $a_j$.
I don't understand statements of (i), (ii) and (iii), which describe something about "period".
(i) What does it mean $P_j=\int_{C_j}f(z)dz$ is the corresponding period of $f(z)$? Does it mean that $f(z+\int_{C_j}f(z)dz)=f(z)$ for all $z$? Why is $f(z+\int_{C_j}f(z)dz)=f(z)$ true?
(ii) Why does $1/(z-a_j)$ have the period $2\pi i$?
(iii) Why does $f(z)-\frac{R_j}{z-a_j}$ have a vanishing period?
