We will briefly discuss zeros and poles of meromorphic functions here. We assume the **Laurent series** exist in the vicinity of a point *z*_{0}:

Clearly if we want *f(z*_{0}*)* is zero we require *a _{k }*is zero for

*k*smaller

**zero. Consequently we define that**

*and equals to**z*

_{0 }is a zero of order

*n*if:

We can observe some useful properties from (2). Firstly, an* n* order zero implies that up to the (*n*-1)^{th }derivative of *f(z) are* also zero at *z*_{0} and vice versa. We can use this property to determine the order of zeros of a function, in the case that they are not so obvious. Secondly, The zeros of *f(z)* are the poles of 1/*f(z)* for obvious reason, provided that *f(z)* is not identically zero. We would like to find the properties of the Laurent series of 1/*f(z)*:

The value of *m* can be determined by multiply the denominator of right hand side : We obtain a series that constantly equals to one. This requires *m*=*n*, and we see the coefficients *b _{k}* are fixed by the values of

*a*.

_{k}In summary we found that

*In particular, if f(z) is analytic and non-zero at z _{0 }, we now from (2) know that n=0 and thus from (4) 1/f(z) is also analytic and non-zero at z_{0.}*