Editing Argument principle (section)

==Formulation==
{{Complex analysis sidebar}}
If ''f'' is a meromorphic function inside and on some closed contour ''C'', and ''f'' has no zeros or poles on ''C'', then 
: <math>\frac{1}{2\pi i}\oint_{C} {f'(z) \over f(z)}\, dz=Z-P</math>
where ''Z'' and ''P'' denote respectively the number of zeros and poles of ''f'' inside the contour ''C'', with each zero and pole counted as many times as its [[Multiplicity (mathematics)|multiplicity]] and [[Pole (complex analysis)|order]], respectively, indicate. This statement of the theorem assumes that the contour ''C'' is simple, that is, without self-intersections, and that it is oriented counter-clockwise.

More generally, suppose that ''f'' is a meromorphic function on an [[open set]] Ω in the [[complex plane]] and that ''C'' is a closed curve in Ω which avoids all zeros and poles of ''f'' and is [[contractible space|contractible]] to a point inside Ω.  For each point ''z'' ∈ Ω, let ''n''(''C'',''z'') be the [[winding number]] of ''C'' around ''z''.  Then
:<math>\frac{1}{2\pi i}\oint_{C} \frac{f'(z)}{f(z)}\, dz = \sum_a n(C,a) - \sum_b n(C,b)\,</math>
where the first summation is over all zeros ''a'' of ''f'' counted with their multiplicities, and the second summation is over the poles ''b'' of ''f'' counted with their orders.