PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] proof of argument principle (Proof)

Since $ f$ is meromorphic, $ f'$ is meromorphic, and hence $ f'/f$ is meromorphic. The singularities of $ f'/f$ can only occur at the zeros and the poles of $ f$.

I claim that all singularities of $ f'/f$ are simple poles. Furthemore, if $ f$ has a zero at some point $ p$, then the residue of the pole at $ p$ is positive and equals the multiplicity of the zero of $ f$ at $ p$. If $ f$ has a pole at some point $ p$, then the residue of the pole at $ p$ is negative and equals minus the multiplicity of the pole of $ f$ at $ p$.

To prove these assertions, write $ f(x) = (x-p)^n g(x)$ with $ g(p) \neq 0$. Then

$\displaystyle {f'(x) \over f(x)} = {n \over x-p} + {g'(x) \over g(x)}$
Since $ g(p) \neq 0$, the second term on the right hand side is not singular at $ p$. The only singularity at $ p$ comes from the first term. Since $ n$ is either the order of the zero of $ f$ at $ p$ if $ f$ has a zero at $ p$ or minus the order of the pole of $ f$ at $ p$ is $ f$ has a pole at $ p$, the assertion is proven.

By the Cauchy residue theorem, the integral

$\displaystyle {1 \over 2 \pi i} \int_C {f'(z) \over f(z)} dz$
equals the sum of the residues of $ f'/f$. Combining this fact with the characterization of the poles of $ f'/f$ and their residues given above, one deduces that this integral equals the number of zeros of $ f$ minus the number of poles of $ f$, counted with multiplicity.



"proof of argument principle" is owned by rspuzio. [ full author list (2) ]
(view preamble | get metadata)

View style:

Other names:  Cauchy's argument principle
Keywords:  complex variables, complex analysis, complex integrals, contour integration, residues

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: number, characterization, sum, integral, Cauchy residue theorem, order, singular, right hand side, term, negative, multiplicity, positive, residue, point, simple poles, poles, meromorphic
There is 1 reference to this entry.

This is version 6 of proof of argument principle, born on 2004-09-04, modified 2006-09-18.
Object id is 6134, canonical name is ProofOfArgumentPrinciple.
Accessed 3576 times total.

Classification:
AMS MSC: 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)