relation between order and multiplicity

Question

Could any one tell me with an example, what is the relation between Order and Multiplicity of a holomorphic function on Riemann Surface,and how this formuale comes?

for example let $p\in X$ be not a pole for $f$, and $f(p)=z_0$ then $f(z)-z_o$ has a simple zero at $p$ so $mult_p(f-f(p))=1$,$ord_p(f-f(p))=ord_p(f)-ord_p(f(p))=1+0?$

Answer 1

Let me just quote Rick Miranda's "Algebraic Curves and Riemann Surfaces":

(Chapter II) Lemma 4.7: Let $f$ be a meromorphic function on a Riemann surface $X$, with associated holomorphic map $F\colon X \to \widehat{\mathbb{C}}$.

a) If $p \in X$ is a zero of $f$, then $\text{mult}_p(F) = \text{ord}_p(f)$.

b) If $p \in X$ is a pole of $f$, then $\text{mult}_p(F) = -\text{ord}_p(f)$.

c) If $p \in X$ neither a zero nor a pole of $f$, then $\text{mult}_p(F) = \text{ord}_p(f - f(p))$.

I don't have time right now, but perhaps later I'll include a proof and an example.