Editing Winding number (section)

===Complex analysis===
Winding numbers play a very important role throughout complex analysis (cf. the statement of the [[residue theorem]]).  In the context of [[complex analysis]], the winding number of a [[closed curve]] <math>\gamma</math> in the [[complex plane]] can be expressed in terms of the complex coordinate {{nowrap|1= ''z'' = ''x'' + ''iy''}}.  Specifically, if we write ''z''&nbsp;=&nbsp;''re''<sup>''iθ''</sup>, then

:<math>dz = e^{i\theta} dr + ire^{i\theta} d\theta</math>

and therefore

:<math>\frac{dz}{z} = \frac{dr}{r} + i\,d\theta = d[ \ln r ] + i\,d\theta.</math>

As <math>\gamma</math> is a closed curve, the total change in <math>\ln (r)</math> is zero, and thus the integral of <math display="inline">\frac{dz}{z}</math> is equal to <math>i</math> multiplied by the total change in <math>\theta</math>.  Therefore, the winding number of closed path <math>\gamma</math> about the origin is given by the expression<ref>{{MathWorld |title=Contour Winding Number |id=ContourWindingNumber |access-date=7 July 2022}}</ref>

:<math>\frac{1}{2\pi i} \oint_\gamma \frac{dz}{z} \, .</math>

More generally, if <math>\gamma</math> is a closed curve parameterized by <math>t\in[\alpha,\beta]</math>, the winding number of <math>\gamma</math> about <math>z_0</math>, also known as the ''index'' of <math>z_0</math> with respect to <math>\gamma</math>, is defined for complex <math>z_0\notin \gamma([\alpha, \beta])</math> as<ref>{{Cite book|url=https://archive.org/details/1979RudinW | title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=0-07-054235-X |pages=201}}</ref>

:<math>\mathrm{Ind}_\gamma(z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{d\zeta}{\zeta - z_0} = \frac{1}{2\pi i} \int_{\alpha}^{\beta} \frac{\gamma'(t)}{\gamma(t) - z_0} dt.</math>

This is a special case of the famous [[Cauchy integral formula]].

Some of the basic properties of the winding number in the complex plane are given by the following theorem:<ref>{{Cite book| url=https://archive.org/details/RudinW.RealAndComplexAnalysis3e1987|title=Real and Complex Analysis|last=Rudin|first=Walter| publisher=McGraw-Hill|year=1987|isbn=0-07-054234-1|edition=3rd |pages=203}}</ref>

'''Theorem.''' ''Let <math>\gamma:[\alpha,\beta]\to\mathbb{C}</math> be a closed path and let <math>\Omega</math> be the set complement of the image of <math>\gamma</math>, that is,  <math>\Omega:=\mathbb{C}\setminus\gamma([\alpha,\beta])</math>.  Then the index of <math>z</math> with respect to <math>\gamma</math>,''<math display="block">\mathrm{Ind}_\gamma:\Omega\to \mathbb{C},\ \ z\mapsto \frac{1}{2\pi i}\oint_\gamma \frac{d\zeta}{\zeta-z},</math>''is (i) integer-valued, i.e., <math>\mathrm{Ind}_\gamma(z)\in\mathbb{Z}</math> for all <math>z\in\Omega</math>; (ii) constant over each component (i.e., maximal connected subset) of <math>\Omega</math>; and (iii) zero if <math>z</math> is in the unbounded component of <math>\Omega</math>.''

As an immediate corollary, this theorem gives the winding number of a circular path <math>\gamma</math> about a point <math>z</math>.  As expected, the winding number counts the number of (counterclockwise) loops <math>\gamma</math> makes around <math>z</math>:

'''Corollary.'''  ''If <math>\gamma</math> is the path defined by <math>\gamma(t)=a+re^{int},\ \ 0\leq t\leq 2\pi, \ \ n\in\mathbb{Z}</math>, then'' <math>\mathrm{Ind}_\gamma(z) = \begin{cases} n, & |z-a|< r; \\ 0, & |z-a|> r. \end{cases}</math>