Argument principle
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. == Formulation == If f is a meromorphic function inside and on some closed contour C, and f has no zeros or poles on C, then 1 2 π i ∮ C f ′ ( z ) f ( z ) d z = Z − P {\displaystyle {\frac {1}{2\pi i}}\oint _{C}{f'(z) \over f(z)}\,dz=Z-P} 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 and order, respectively, indicate.