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 by a parameter \( t, \), then \( N\left(\mathcal{C}_{t} ; a\right) \) would vary continuously with \( t . \quad \) Finally, when \( \mathrm{C} \) is closed, \( N \) takes on the discrete values 0,\( \pm 1, \pm 2, . . . . \) Thus the winding number for a closed path $\mathrm C$ about a point  \( a \) remains constant, as \( \mathrm{C} \) and  \( a \) vary continuously, as long as  \( a \) is never on \( \mathrm{C} \).

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