The Argument Principle states that for a function meromorphic inside and on a simple closed contour, with no zeros or poles on the contour itself, the number of zeros minus the number of poles enclosed, each counted with multiplicity, equals the number of times the function's value winds around the origin as the contour is traversed, computed as a contour integral of the function's logarithmic derivative. It is a standard tool of complex analysis for counting zeros and poles without finding them explicitly, and it underlies the proof of the already-live Rouche's Theorem on this atlas.
Facts
StatementFor a function meromorphic inside and on a simple closed contour, with no zeros or poles on the contour itself, the number of zeros minus the number of poles enclosed, each counted with multiplicity, equals the winding number of the image contour around the origin, computed as a contour integral of the function's logarithmic derivative. 1 Proof YearCauchy presented a version covering zeros only in 1831, published only in 1874 in hand-written form; 1855 is when he published a paper discussing both zeros and poles, matching this entity's own statement. Classification
Statement Form Connections
Sources
1. Argument Principle (Wikipedia)
Wikimedia Foundationlead paragraph, definition
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
History section
Cauchy published a paper with a discussion on both zeroes and poles in 1855, two years before his death.
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