# Ngô Quốc Anh

## July 15, 2010

### The winding number is a special case of the Brouwer degree

Filed under: Giải tích 7 (MA4247) — Ngô Quốc Anh @ 18:25

In the classical complex analysis, the winding number of a plane curve $\Gamma$ with respect to a point $a \notin \mathbb C \setminus \Gamma$ is defined by

$\displaystyle w(\Gamma ,a) = \frac{1}{{2\pi i}}\int_\Gamma {\frac{{dz}}{{z - a}}}$.

Let $\Omega \subset \mathbb R^n$ be open and bounded. Let $f : \Omega \to \mathbb R^n$ and $p \notin f(\partial \Omega)$. Notation $J_f(a)$ denotes the Jacobian of $f$ evaluated at $a$. It is well-known that the Brouwer Degree Theory, usually denoted by $\deg$, is constructed for continuous function, $C^0(\overline\Omega)$-class, via the following steps

For the $C^1(\overline\Omega)$-class: We assume $J_f(f^{-1}(p)) \ne 0$ then

$\displaystyle\deg (f,\Omega ,p) = \begin{cases} \sum\limits_{x \in {f^{ - 1}}(p)} {{\rm sgn} {J_f}(x)} , & {f^{ - 1}}(p) \ne \emptyset ,\\ 0, & {f^{ - 1}}(p) = \emptyset.\end{cases}$

For the $C^2(\overline\Omega)$-class: In case we want to remove the condition $J_f(f^{-1}(p)) \ne 0$, we then define

$\displaystyle\deg (f,\Omega ,p) = \deg (f,\Omega ,p')$

where $p'$ is any regular value of $f$ sufficiently closed to $p$ in the sense that

$\displaystyle\left\| {p - p'} \right\| < {\rm dist}(p,f(\partial \Omega ))$.

For the $C^0(\overline\Omega)$-class: In this case, we define

$\displaystyle\deg (f,\Omega ,p) = \deg (g,\Omega ,p)$

where $g \in C^2(\overline\Omega)$ is sufficiently closed to $f$ in the sense that

$\displaystyle\left\| f-g \right\| < {\rm dist}(p,f(\partial \Omega ))$.

Now we prove the following fact

Theorem.  Let $B(0,1)\subset \mathbb C$ be the unit ball and $\Gamma = \partial B(0,1)$. Assume $f : \overline{B(0,1)} \to \mathbb C$ is a $C^1$ function and $a \notin f(\Gamma)$. Then

$\displaystyle\deg (f,B(0,1),a) = \frac{1}{{2\pi i}}\int_{f(\Gamma )} {\frac{{dz}}{{z - a}}}$.