Embracing Analytically and Winding
Complex Logarithm and Winding For any path \( \mathcal{C} \) and any point \( a \) not on \( \mathrm {C}, \) the complex logarithm enables us to define the following function: \[\begin{array}{l} \quad N(\mathcal{C} ; a)=\frac{1}{2 \pi i} \int_{\mathrm{C}} \frac{d z}{z-a}=\frac{1}{2 \pi i} \int_{z} d \log (z-a)=\left.\frac{1}{2 \pi} \arg (z-a)\right|_{\mathrm C} \end{array}\] This is called the winding number of the path \( \mathrm{C} \) with respect to \( a \). Since the indeterminacy of the argument is in multiples of \( 2 \pi \), if \( \mathrm{C} \) is a closed path, then the values of \( N(\mathrm{C} ; a) \) are integers. The expression \( N(\mathrm{C} ; a) \) is simultaneously a function of both \( \mathrm{C} \) and \( a \). Evidently, \( N \) depends continuously on \( a \) and $\mathrm C$ unless \( a \) reaches \( \mathrm{C} . \) This means that if a family of paths \( \mathcal{C}_{t} \), where none of them is passing through \( a \), were parametrized continuously b...