Let \(f\) be holomorphic on a punctured disc \(D_r(z_0)\setminus\{z_0\}\). The point \(z_0\) is a pole of \(f\) if there is \(0<\rho\le r\) such that \(f\) has no zeros on \(D_\rho(z_0)\setminus\{z_0\}\) and \(g\coloneqq1/f\), extended by \(g(z_0)\coloneqq0\), is holomorphic on \(D_\rho(z_0)\). The order of the pole is the least integer \(n\ge1\) for which there is a function \(h\) holomorphic on a neighborhood of \(z_0\) with \[ f(z)=(z-z_0)^{-n}h(z) \] for \(z\neq z_0\) near \(z_0\). A pole of order \(1\) is a simple pole.
Let \(f\) be holomorphic on \(D_r(z_0)\setminus\{z_0\}\).
This is the last assertion of the proposition on D-AUD6K, whose condition (b) is the present definition.
Since \(g=1/f\) is holomorphic near \(z_0\), not identically zero, and \(g(z_0)=0\), it has a zero of some order \(n\ge1\): \(g(z)=(z-z_0)^nk(z)\) with \(k(z_0)\neq0\), and \(h_0\coloneqq1/k\) is holomorphic near \(z_0\) with \(f(z)=(z-z_0)^{-n}h_0(z)\) and \(h_0(z_0)\neq0\). If \(f(z)=(z-z_0)^{-m}h(z)\) with \(h\) holomorphic near \(z_0\) and \(m<n\), then \(h(z)=(z-z_0)^{m-n}h_0(z)\) for \(z\neq z_0\), which is unbounded near \(z_0\), a contradiction. So \(n\) is the least such integer.