Zeros and poles

Theorem 11.1 (Zeros and poles). Let \(m \geq 1\).

(i)

\(f\) has a pole of order \(m\) at \(z_0\) if and only if \(f(z) = g(z) / (z - z_0)^m\) near \(z_0\), with \(g\) analytic at \(z_0\) and \(g(z_0) \neq 0\).

(ii)

\(g\) has a zero of order \(m\) at \(z_0\) if and only if \(g(z) = (z - z_0)^m h(z)\) with \(h\) analytic at \(z_0\) and \(h(z_0) \neq 0\).

(iii)

If \(g\) has a zero of order \(m\) at \(z_0\), then \(1/g\) has a pole of order \(m\) there.

Source: [1, ch. 5]