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Argument Principle

If f(z) is meromorphic in a region R enclosed by a contour gamma, let N be the number of complex roots of f(z) in gamma, and P be the number of poles in gamma, with each zero and pole counted as many times as its multiplicity and order, respectively. Then

N-P=1/(2pii)int_gamma(f^'(z)dz)/(f(z)).

Defining w=f(z) and sigma=f(gamma) gives

N-P=1/(2pii)int_sigma(dw)/w.

See also

Cauchy Integral Formula, Cauchy Integral Theorem, Hurwitz's Root Theorem, Meromorphic Function, Pole, Root, Rouché's Theorem, Variation of Argument

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References

Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Argument Principle for Harmonic Functions." Amer. Math. Monthly 103, 411-415, 1996.Knopp, K. Theory of Functions, Parts I and II. New York: Dover, pp. 132-134, 1996.Krantz, S. G. "The Argument Principle." Ch. 5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 69-78, 1999.

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Argument Principle

Cite this as:

Weisstein, Eric W. "Argument Principle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ArgumentPrinciple.html

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