Let \(\gamma\) be a positively oriented simple closed piecewise smooth curve, and let \(f\) be meromorphic on an open set containing \(\gamma\) and its interior, with no zeros or poles on \(\gamma\). At a zero of order \(m\) the logarithmic derivative \(f'/f\) has a simple pole with residue \(m\), and at a pole of order \(m\) a simple pole with residue \(-m\), so the argument principle gives \(\frac{1}{2\pi i}\oint_\gamma\frac{f'}{f}\,dz=Z-P\), the numbers of zeros and poles inside \(\gamma\) counted with multiplicity. The same integral is the winding number of the curve \(f\circ\gamma\) about \(0\). Rouché’s theorem: if \(f\) and \(g\) are holomorphic on an open set containing \(\gamma\) and its interior and \(\abs{g}<\abs{f}\) on \(\gamma\), then \(f\) and \(f+g\) have the same number of zeros inside \(\gamma\). Applying the argument principle to \(f-w\) for \(w\) near \(f(z_0)\) proves the open mapping theorem: a nonconstant holomorphic function on a domain is an open map. The maximum modulus principle follows again, and applying it to \(1/f\) gives the minimum modulus principle for nonvanishing \(f\).
Logarithmic derivative (Tag D-IA5FS)
Let \(\Omega\subseteq\CC\) be open and let \(f\) be meromorphic on \(\Omega\) and not identically zero on any connected component of \(\Omega\). The logarithmic derivative of \(f\) is the meromorphic function on \(\Omega\) \[ \partial_{\log}f\coloneqq\frac{f'}{f}. \]
Let \(f\) be as in the definition and let \(z_0\in\Omega\) be a zero of order $m$ or a pole of order $m$ of \(f\). Then \(f'/f\) has a simple pole at \(z_0\) with residue \(m\) in the first case and \(-m\) in the second. At every other point of \(\Omega\), \(f'/f\) is holomorphic.
Near \(z_0\) write \(f(z)=(z-z_0)^{k}h(z)\) with \(h\) holomorphic near \(z_0\) and \(h(z_0)\neq0\), where \(k=m\) at a zero and \(k=-m\) at a pole. Then, for \(z\neq z_0\) near \(z_0\), \[ \frac{f'(z)}{f(z)}=\frac{k}{z-z_0}+\frac{h'(z)}{h(z)}, \] and \(h'/h\) is holomorphic near \(z_0\). At a point where \(f\) is holomorphic and nonzero, \(f'/f\) is holomorphic.
Winding number (Tag D-PJ7JM)
Let \(\gamma\) be a closed piecewise \(C^1\) curve in \(\CC\) and let \(z_0\in\CC\) be a point not on \(\gamma\). The winding number, or index, of \(\gamma\) about \(z_0\) is \[ n(\gamma, z_0)\coloneqq\frac{1}{2\pi i}\int_\gamma\frac{1}{\xi-z_0}\dxi. \]
Argument principle counting zeros and poles (Tag T-VCX3Y)
Let \(\Omega\subseteq\CC\) be a bounded open set whose boundary \(\gamma\coloneqq\partial\Omega\) consists of finitely many piecewise smooth closed curves, oriented positively with respect to \(\Omega\). Let \(f\) be meromorphic on an open set containing \(\overline{\Omega}\), with no zeros or poles on \(\gamma\), and let \(Z_f\) and \(P_f\) be the numbers of zeros and poles of \(f\) in \(\Omega\), counted with multiplicity. Then \[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz=Z_f-P_f. \]
Argument principle as a winding number (Tag T-EBAOE)
Let \(\Omega\subseteq\CC\) be open, let \(f\) be meromorphic on \(\Omega\), and let \(\gamma\colon[a,b]\to\Omega\) be a piecewise smooth closed curve on which \(f\) has no zeros or poles. Then \[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz=\Index_{w=0}(f\circ\gamma), \] the winding number of the closed curve \(f\circ\gamma\) about \(0\).
The curve \(f\circ\gamma\) is closed, piecewise smooth, and avoids \(0\). Substituting \(w=f(\gamma(t))\), so that \(\dw=f'(\gamma(t))\gamma'(t)\dt\), gives \[ \Index_{w=0}(f\circ\gamma)=\frac{1}{2\pi i}\int_{f\circ\gamma}\frac{\dw}{w}=\frac{1}{2\pi i}\int_a^b\frac{f'(\gamma(t))}{f(\gamma(t))}\gamma'(t)\dt=\frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz. \]
Rouché's theorem (Tag T-5JNUU)
Let \(\Omega\subseteq\CC\) be open, let \(f,g\) be meromorphic on \(\Omega\), and let \(\gamma\) be a positively oriented toy contour such that \(\gamma\) and its interior lie in \(\Omega\). Suppose that \(f\) and \(f+g\) have no zeros or poles on \(\gamma\), and that \[ \abs{g(z)}<\abs{f(z)}\qquad\text{for all }z\in\gamma. \] Let \(Z_h\) and \(P_h\) be the numbers of zeros and poles of \(h\) inside \(\gamma\), counted with multiplicity. Then \(\Index_{w=0}(f\circ\gamma)=\Index_{w=0}\big((f+g)\circ\gamma\big)\), and hence \[ Z_f-P_f=Z_{f+g}-P_{f+g}. \] In particular, if \(f\) and \(g\) are holomorphic on \(\Omega\), then \(f\) and \(f+g\) have the same number of zeros inside \(\gamma\).
On \(\gamma\), \(\frac{f+g}{f}=1+\frac gf\) takes values in the disc \(\ts{\abs{w-1}<1}\), which does not contain \(0\), so the closed curve \(\frac{f+g}{f}\circ\gamma\) has winding number \(0\) about \(0\). The logarithmic derivative of a product is the sum of the logarithmic derivatives, so by T-52HK6 \[ \Index_{w=0}\big((f+g)\circ\gamma\big)=\Index_{w=0}(f\circ\gamma)+\Index_{w=0}\Big(\frac{f+g}{f}\circ\gamma\Big)=\Index_{w=0}(f\circ\gamma). \] By the argument principle, \(\Index_{w=0}(h\circ\gamma)=Z_h-P_h\) for \(h=f\) and \(h=f+g\).
Open mapping theorem (Tag C-FRF33)
Let \(\Omega\subseteq\CC\) be a connected open set and let \(f\) be a nonconstant holomorphic function on \(\Omega\). Then \(f\) is an open map: \(f(U)\) is open for every open set \(U\subseteq\Omega\).
Minimum modulus principle (Tag T-YLI6Y)
Let \(\Omega\subseteq\CC\) be a connected open set and let \(f\) be holomorphic and nonvanishing on \(\Omega\).
If \(\abs{f}\) has a local minimum at some \(z_0\in\Omega\), then \(f\) is constant on \(\Omega\).
If moreover \(\Omega\) is bounded, \(f\) extends continuously to \(\overline{\Omega}\), and \(f\) is nonconstant, then \(\min_{\overline{\Omega}}\abs{f}\) is attained only at points of \(\partial\Omega\).