# argument principle

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

$\frac{1}{2\pi i}}{\displaystyle {\oint}_{C}}{\displaystyle \frac{{f}^{\prime}(z)}{f(z)}}\mathit{d}z$ | (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 ${f}^{\prime}/f={(\mathrm{log}f)}^{\prime}$ and that $\mathrm{log}(z)=\mathrm{log}|z|+i\mathrm{arg}z$. Substituting this into formula^{} (1), we find

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

The first integral on the rightmost side of this equation equals zero because $\mathrm{log}|f|$ 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 $f\circ C$.

Title | argument principle |
---|---|

Canonical name | ArgumentPrinciple |

Date of creation | 2013-03-22 14:34:28 |

Last modified on | 2013-03-22 14:34:28 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Algorithm |

Classification | msc 30E20 |

Synonym | Cauchy’s argument principle |

Defines | argument principle |