argument principle

If a function $f$ is meromorphic on the interior of a rectifiable simple closed curve $C$, then

${\displaystyle \frac{1}{2\pi i}}{\displaystyle {\oint }_C}{\displaystyle \frac{f^{\prime }(z)}{f(z)}}𝑑z$(1)

equals the difference between the number of zeros and the number of poles of $f$ counted with multiplicity. (For example, a zero of order two counts as two zeros; a pole of order three counts as three poles.)This fact is known as the argument principle.

The principle may be stated in another form which makes the origin of the name apparent: If a function $f$ is meromorphic on the interior of a rectifiable simple closed curve $C$ and has $m$ poles and $n$ zeros on the interior of $C$, then the argument of $f$ increases by $2\pi (n-m)$ upon traversing $C$. The relation of this statement to the previous statement is easy to see. Note that $f^{\prime }/f={(\log f)}^{\prime }$ and that $\log (z)=\log |z|+i\arg z$. Substituting this into formula (1), we find

$$2\pi i(n-m)={\oint }_C\frac{f^{\prime }(z)}{f(z)}𝑑z={\oint }_{C}d\log |f(z)|+i{\oint }_{C}d\arg (f(z)).$$

The first integral on the rightmost side of this equation equals zero because $\log |f|$ is single-valued. The second integral on the rightmost side equals the change in the argument as one traverses $C$. Cancelling the $i$ from both sides, we conclude that the change in the argument equals $2\pi (n-m)$.

Note also that the integral (1)is the winding number, about zero, of the image curve $f\circ C$.