Argument Principle

If f(z) is meromorphic in a region R enclosed by a contour gamma, let N be the number of complex roots of f(z) in gamma, and P be the number of poles in gamma, with each zero and pole counted as many times as its multiplicity and order, respectively. Then

 N-P=1/(2pii)int_gamma(f^'(z)dz)/(f(z)).

Defining w=f(z) and sigma=f(gamma) gives

 N-P=1/(2pii)int_sigma(dw)/w.

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