The zeros of a holomorphic function in a region can be counted by factoring, by Rouché’s theorem, by the argument principle, or, for a locally uniform limit, by Hurwitz’s theorem.
Explicit factorization
For a polynomial whose roots can be computed, the factorization gives the zeros and their multiplicities.
A zero of \(f\) of multiplicity \(m \geq 2\) is a zero of \(f'\), and a solution of \(f(z) = a\) of multiplicity \(m\geq 2\) is a zero of \(f'\).
\(f\) and \(f-w\) have the same derivative, so the multiple solutions of \(f(z)=w\), for every \(w\), lie among the zeros of \(f'\).
A dominant term on the boundary
Rouché’s theorem. If \(f = M + m\) with \(\abs{m} < \abs{M}\) on the boundary curve \(\gamma\), then \(f\) and \(M\) have the same number of zeros minus poles inside \(\gamma\) (Rouché's theorem). This applies when \(M\) is a term whose zeros inside \(\gamma\) are known, such as one monomial of a polynomial that dominates on \(\abs z = R\), or an entire function compared with a polynomial. The dominant term depends on the curve: for \(z^4+6z+3\), it is \(z^4\) on \(\abs z = 2\) and \(6z\) on \(\abs z = 1\) (Rouché's theorem).
(Rouché's theorem fails with $\abs m\leq\abs M$ on $\gamma$)
On \(\gamma = \ts{\abs z = 1}\), take \(M(z) \coloneqq z\) and \(m(z) \coloneqq -1\), so \(\abs m \leq \abs M\) on \(\gamma\) with equality everywhere. Then \(M\) has one zero in \(\DD\), while \(M + m = z - 1\) has none in \(\DD\); its zero lies on \(\gamma\).
The image of the boundary curve
Argument principle. For \(f\) meromorphic with no zeros or poles on \(\gamma\), the number of zeros minus poles inside \(\gamma\) is the winding number of \(f\circ\gamma\) about \(0\) (Argument principle as a winding number). It applies when the image curve \(f\circ\gamma\) or the change of \(\arg f\) along \(\gamma\) is known, and it counts zeros and poles together.
A locally uniform limit
Hurwitz’s theorem. If holomorphic \(f_k \to f\) locally uniformly on a connected open set, then near a zero of order \(n\) of \(f\), the functions \(f_k\) have exactly \(n\) zeros for large \(k\) (Hurwitz's theorem). Consequently a limit of nowhere-zero functions is nowhere zero or identically zero, and a limit of univalent functions is univalent or constant (Locally uniform limits of univalent functions are univalent or constant).
Solutions of \(f(z) = w\)
Rouché’s theorem applies to \(f - w\). The argument principle expresses the number of solutions in \(\Omega\) as \[ F(w) \coloneqq {1\over 2\pi i} \oint_{\bd \Omega} {f'(z) \over f(z) - w} \dz ,\] which is continuous and integer valued on each connected component of \(\CC\sm f(\bd\Omega)\), hence constant there. So the number of solutions of \(f(z)=w\) in \(\Omega\) is the same for all \(w\) in one component of \(\CC\sm f(\bd\Omega)\).
Comparison
| Argument principle | Rouché | |
|---|---|---|
| Input | \(f\circ\gamma\), or the change of \(\arg f\) along \(\gamma\) | a decomposition \(f=M+m\) with \(\abs m<\abs M\) on \(\gamma\) |
| Conclusion | \(Z_f - P_f\) equals a winding number | \(Z_f - P_f = Z_M - P_M\) |
Rouché’s theorem is proved from the argument principle.