# Statement:

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

(1/2πi) \oint_{γ} 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,

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 \oint

_{γ} f'(z)/f(z) dz = \oint

_{f o γ} 1/u du, and the definition of the winding number.

# Interpretation:

Personally, I think this ranks as one of the most beautiful mathematical results since

e^{iπ}+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).