Quick question on the roots and poles of a meromorphic function,

Question

Does the degree of the polynomial in the numerator always equal the degree of the polynomial in the denominator?

In other words, the number of zeros, counting multiplicity, equals the number of poles, counting multiplicity, of a meromorphic function.

So something like

$$\frac{z^2}{(z-1)(z+1)}$$

would be a legitimate construction of a meromorphic function

while

$$\frac{z^5}{(z-1)(z+1)}$$

is not a meromorphic function, since the degrees of its zeros and poles do not match. It is simply a rational function.

Is my thinking correct?

Thanks,

Answer 1

You are forgetting $\infty$. The second example has a pole of order 3 at the point at infinity.