representation for a pole of order N

Question

If a meromorphic function has a single pole of order $N$ at $z_0$, then is the principal part of that function always

$$\frac{c}{(z-z_0)^N}$$ for some constant $c$? A counterexample or proof would be really helpful. Thanks.

Answer 1

No, of course not. Take$$\frac1{z^2}+\frac1z\tag1$$(with domain equal to $\mathbb{C}\setminus\{0\}$) for instance. The principal part of this function is $(1)$.

Answer 2

No, the principal part will be $$ \frac{c_N}{(z-z_0)^N}+\frac{c_{N-1}}{(z-z_0)^{N-1}}+\dots+\frac{c_1}{z-z_0}. $$