Theorem: Given a meromorphic function f and a closed contour γ, with no zeroes or poles of f lying on γ,

      (1/2πi) \ointγ f'(z)/f(z) dz = #(zeroes of f inside γ) - #(poles of f inside γ), counted to multiplicity.

    and is denoted ZP(f,γ). Furthermore,

      ZP(f,γ) = w(f o γ,0)

    ie. is the winding number of f o γ about zero.
The counted to multiplicity part means that a zero of order two adds two to the count, a pole of order three subtracts three from the count, etc. The \oint means a contour intergral, in case you thought it looked a bit odd.

Sketch of Proof:

Fairly straightforward: for the first equation, note that the only singularities of the integrand are when f has a zero or a pole, then use the Residue Theorem on f'/f, using the Taylor Expansions of f and f' near the zeroes and poles. The second equation comes from noting that \ointγ f'(z)/f(z) dz = \ointf o γ 1/u du, and the definition of the winding number.


Personally, I think this ranks as one of the most beautiful mathematical results since e+1 = 0. As for why, I can't really say (you have to see it for yourself...), except that it provides the most elegant and concise way of determining where the zeroes of an analytic function lie (not their precise location, that would be too good to ask for): just apply f to the contour and see how many times the resulting curve winds around zero (you do this by looking at the argument of f o γ hence the name). Indeed, the theorem can be used to prove the Fundamental Theorem of Algebra (by showing that for a not-constant polynomial p, p o γ must wind around zero when γ is sufficiently large).

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