If f(z) is meromorphic in a region R enclosed by a contour γ, let N be the number of complex roots of f(z) in γ, and P be the number of poles in γ, with each zero and pole counted as many times as its multiplicity and order, respectively. Then N - P = 1/(2π i) integral_γ (f'(z) d z)/(f(z)). Defining w congruent f(z) and σ congruent f(γ) gives N - P = 1/(2π i) integral_σ (d w)/w.

Cauchy integral formula | Cauchy integral theorem | Hurwitz's root theorem | meromorphic function | pole | root | Rouché's theorem | variation of argument