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# argument principle

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

$\displaystyle{1\over 2\pi i}\oint_{C}{f^{{\prime}}(z)\over f(z)}dz$ | (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

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

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 $f\circ C$.

## Mathematics Subject Classification

30E20*no label found*

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