The winding number is the increment of a continuous argument divided by $2\pi$

Given: A closed complex contour $\gamma :[a,b]\to \mathbb{C}$, a point $p\mathrm{\notin }{\gamma }^{\ast }$, a continuous logarithm $\lambda$ of $\gamma -p$ along $\gamma$, and $\theta =\mathrm{Im}\lambda$.

For a closed complex contour and $p$ off its trace, $n(\gamma ,p)\in \mathbb{Z}$ (The winding number of a closed contour is an integer), where $n(\gamma ,p)=(2\pi i{)}^{-1}{\int }_{\gamma }dz\mathrm{/}(z-p)$ (The winding number of a closed contour about a point off its trace).

A continuous logarithm of $\gamma -p$ along $\gamma$ is a continuous $\lambda$ with $\exp (\lambda (t))=\gamma (t)-p$ for every $t$, its continuous argument is $\theta =\mathrm{Im}\lambda$, and any two continuous logarithms differ by a constant in $2\pi i\mathbb{Z}$ (Continuous logarithms and continuous arguments along a contour, Every contour missing a point admits a continuous logarithm, unique up to a constant in $2\pi i\mathbb{Z}$).

For real $x,y$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$ ($\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$).

For a complex contour $\gamma$, $p\mathrm{\notin }{\gamma }^{\ast }$ and a continuous logarithm $\lambda$ of $\gamma -p$ along $\gamma$, ${\int }_{\gamma }dz\mathrm{/}(z-p)=\lambda (b)-\lambda (a)$ (The integral of $dz\mathrm{/}(z-p)$ along a contour is the increment of a continuous logarithm).

For $x>0$, $\log x$ is the unique real $y$ with $\exp y=x$ (The natural logarithm as the inverse of the exponential function).

For $z=a+bi$ with $a,b$ real, $\mathrm{Re}z=a$ and $\mathrm{Im}z=b$ (Real and imaginary parts, complex conjugation, and modulus).

A complex contour is closed when $\gamma (a)=\gamma (b)$ (Rectifiable complex contours, reversal, concatenation, closedness, and orientation).

The integers form a commutative ring (The integers form a commutative ring).