Argument principle
Theorem. If \(f\) is meromorphic inside and on a contour \(\gamma\), with no zeros or poles on \(\gamma\), then \[ \frac{1}{2\pi i}\int_\gamma \frac{f'(z)}{f(z)}\,\mathrm{d}z=N-P, \] where zeros and poles inside are counted with multiplicity.
Proof through logarithmic derivative
Proof. Near a zero or pole of order \(m\), write \(f(z)=(z-a)^m g(z)\) with \(g(a)\ne0\), allowing \(m<0\) for poles. Then \(f'/f=m/(z-a)+g'/g\), so the residue of \(f'/f\) is \(m\).
Rouché theorem
Theorem. If \(f\) and \(g\) are holomorphic inside and on \(\gamma\) and \(|g|<|f|\) on \(\gamma\), then \(f\) and \(f+g\) have the same number of zeros inside \(\gamma\), counted with multiplicity.
Homotopy proof of Rouché
Proof. For \(0\le t\le1\), the functions \(f+tg\) do not vanish on \(\gamma\). Their winding numbers around zero are constant in \(t\), so the argument principle gives the same zero count for \(f\) and \(f+g\).
Polynomial example
Example. On \(|z|=2\), the term \(z^5\) dominates \(3z^2+1\). Hence \(z^5+3z^2+1\) has five zeros in \(|z|<2\). On smaller contours, a different dominant term may control a local zero count.
Winding interpretation
Remark. The argument principle says that zeros and poles measure how the image curve winds around the origin. Rouché's theorem is stability of that winding number under boundary-small perturbations.
Logarithmic derivatives count divisors
Theorem (argument principle). Let \(f\) be meromorphic inside a cycle \(\Gamma\), with no zeros or poles on \(\Gamma\). Then \[ \frac1{2\pi i}\int_\Gamma\frac{f'(z)}{f(z)}\,\mathrm{d}z =\sum_a\operatorname{Ind}(\Gamma,a) \operatorname{ord}_a(f). \] For a simple positively oriented boundary this is the number of zeros minus poles, counted with multiplicity.
Proof. Near \(a\), \(f(z)=(z-a)^mg(z)\) with \(g(a)\ne0\), where \(m\) is positive for a zero and negative for a pole. Then \[ \frac{f'}f=\frac m{z-a}+\frac{g'}g, \] so its residue is \(m\). Apply the residue theorem.
The integral is the net change of a continuous argument of \(f(\Gamma)\) divided by \(2\pi\). Thus the theorem is also a winding-number statement: zeros inside force the image contour to wind around the origin.
Rouché's theorem
Theorem (Rouché). If \(f,g\) are holomorphic inside and on a simple closed contour \(\Gamma\), and \[ |g(z)|<|f(z)|\quad(z\in\Gamma), \] then \(f\) and \(f+g\) have the same number of zeros inside, counted with multiplicity.
Proof. For \(0\le t\le1\), \[ f_t=f+tg \] has no boundary zero because \(|tg|<|f|\). The integer \[ N(t)=\frac1{2\pi i}\int_\Gamma\frac{f_t'}{f_t}\,\mathrm{d}z \] depends continuously on \(t\), hence is constant. Its endpoint values count the zeros of \(f\) and \(f+g\).
Example. On \(|z|=2\), \[ |3z^4|=48>|2z^2-z+1|\le11. \] Therefore \(3z^4+2z^2-z+1\) has four zeros in \(|z|<2\). Rouché counts them without solving the quartic.
Local mapping degree
If \(f(z)-w_0\) has a zero of order \(m\) at \(z_0\), choose a small disc containing no other preimage of \(w_0\). Rouché shows that every nearby \(w\ne w_0\) has exactly \(m\) preimages in the disc, counted with multiplicity. This is the analytic local degree and explains branch points.