21 - Argument Principle

Argument Principle

Suppose $f$ is meromorphic on $U$
Let $\gamma$ be simply positively oriented contour such that $\gamma$ and it’s inside is contained in $U$

Assume $f$ has no zeroes or poles on ${\gamma }^{\ast }$

Let $N$ is the number of zeros (counted with multiplicity) of $f$ inside $\gamma$
Let $P$ be the number of poles (counted with multiplicity) of $f$ inside $\gamma$ then

$$N-P=\frac{1}{2\pi i}{\int }_{\gamma }\frac{f^{\prime }(z)}{f(z)}dz$$

Moreover it is also the winding number of path $\Gamma =f\circ \gamma$ about the origin

Proof

Curve is simple so winding number around any point inside is $1$
By Residue of the logarithmic derivative then
$\frac{f^{\prime }(z)}{f(z)}$ has simple poles at zeros and poles of $f$ with residues given by their order
So result follows from Residue Theorem

For the last part then

$${\int }_{f\circ \gamma }\frac{1}{w}dw={\int }_0^1\frac{1}{f(\gamma (t))}{f}^{\prime }(\gamma (t)){\gamma }^{\prime }(t)dt={\int }_{\gamma }\frac{f^{\prime }(z)}{f(z)}dz$$