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Argument Principle

If $f(z)$ is Meromorphic in a region $R$ enclosed by a curve $\gamma$, let $N$ be the number of Complex Roots of $f(z)$ in $\gamma$, and $P$ be the number of Poles in $\gamma$, then

N-P = {1\over 2\pi i} \int_\gamma {f'(z)\,dz\over f(z)}.

Defining $w \equiv f(z)$ and $\sigma\equiv f(\gamma)$ gives

N-P = {1\over 2\pi i} \int_\sigma {dw\over w}.

See also Variation of Argument


Duren, P.; Hengartner, W.; and Laugessen, R. S. ``The Argument Principle for Harmonic Functions.'' Math. Mag. 103, 411-415, 1996.

© 1996-9 Eric W. Weisstein