Principle of the argument

The principle of the argument is a theorem from function theory , which expresses the poles and places of a meromorphic function counted with multiples by an integral. ${\displaystyle w}$

statement

Be open and cohesive. Be a meromorphic function such that . Be , the amount of -Make and the amount of the poles of . Be and the respective multiplicities. Let be a zero homologous cycle located in such that . Then follows ${\displaystyle \Omega \subseteq \mathbb {C} }$ ${\displaystyle f\colon \Omega \rightarrow {\overline {\mathbb {C} }}}$ ${\displaystyle f\neq 0}$ ${\displaystyle w\in \mathbb {C} }$ ${\displaystyle N=\{z\in \Omega \mid f(z)=w\}}$ ${\displaystyle w}$ ${\displaystyle P=\{z\in \Omega \mid f(z)=\infty \}}$ ${\displaystyle f}$ ${\displaystyle n\in \mathbb {N} ^{N}}$ ${\displaystyle m\in \mathbb {N} ^{P}}$ ${\displaystyle \gamma }$ ${\displaystyle \Omega }$ ${\displaystyle \forall z\in N\cup P:z\notin \mathrm {Bild} (\gamma )}$

${\displaystyle {\frac {1}{2\pi i}}\int _{\gamma }{\frac {f'(z)}{f(z)-w}}\mathrm {d} z=\sum _{z\in N}{\operatorname {ind} _{\gamma }(z)n(z)}-\sum _{z\in P}{\operatorname {ind} _{\gamma }(z)m(z)}}$,

where the number of revolutions of the cycle denotes um . ${\displaystyle \operatorname {ind} _{\gamma }(z)}$ ${\displaystyle \gamma }$ ${\displaystyle z}$

Remarks

The principle of the argument is a simple consequence of the residual theorem . The Rouché Theorem can be derived as an application .

Individual evidence

1. ↑ Dietmar A. Salamon : Function Theory , Springer, Basel 2012, ISBN 9783034801683 , Chapter 4.5: The principle of the argument .
2. ↑ Wolfgang Fischer, Ingo Lieb: Function theory . Vieweg-Verlag 1980, ISBN 3-528-07247-4 , Chapter IV Isolated Singularities , Sentence 7.1