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'argument principle'
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| Title of object: |
argument principle |
| Canonical Name: |
ArgumentPrinciple |
| Type: |
Algorithm |
| Created on: |
2004-09-04 01:33:32 |
| Modified on: |
2006-12-23 09:35:47 |
| Classification: |
msc:30E20 |
| Keywords: |
argument, complex anaysis, contour integration |
| Defines: |
argument principle |
| Synonyms: |
argument principle=Cauchy's argument principle |
Revision comment (for changes between this and next version):
| fixed typos (and suppressed wrong links while I'm at it) |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
If a function $f$ is meromorphic on the interior of a rectifiable simple closed curve $C$, then
$${1 \over 2 \pi} \oint_C {f'(z) \over f(z)} dz$$
equals the difference between the number of zeros and the number of poles of $f$ counted with multiplicity. (e.g. a second order zero counts as two zeros, a third order pole counts as three poles)
The argument 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 $f'/f = (log f)'$ and that $f\log (z) = \log |z| + i \arg z$. Substitituting this into the formula of last paragraph,
$$2 \pi i (n - m) = \oint_C {f'(z) \over f(z)} dz = \oint_C d \log |f(z)| + i \oint_C d \arg (f(z))$$
The first integral on the rightmost side of the 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 $2 \pi (n - m)$. |
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