The logarithmic derivative counts how many times $f\circ\gamma$ winds around $0$.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open, let \(f\) be meromorphic on \(\Omega\), and let \(\gamma\colon[a,b]\to\Omega\) be a closed piecewise \(C^1\) curve that meets no zero and no pole of \(f\). Then the integral of the logarithmic derivative of \(f\) along \(\gamma\) is the winding number of the closed curve \(f\circ\gamma\) about \(0\): \[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz=\Index_{w=0}(f\circ\gamma)=\frac{1}{2\pi i}\int_{f\circ\gamma}\frac{dw}{w}. \]
The curve \(f\circ\gamma\) is closed, piecewise \(C^1\), and avoids \(0\). Substituting \(w=f(\gamma(t))\), \(dw=f'(\gamma(t))\gamma'(t)\dt\), \[ \int_\gamma\frac{f'(z)}{f(z)}\dz=\int_a^b\frac{f'(\gamma(t))\gamma'(t)}{f(\gamma(t))}\dt=\int_{f\circ\gamma}\frac{dw}{w}. \]
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