argument variation

Let [arg(f(z))] denote the change in the complex argument of a function f(z) around a contour γ. Also let N denote the number of roots of f(z) in γ and P denote the sum of the orders of all poles of f(z) lying inside γ. Then [arg(f(z))] = 2π(N - P). For example, the plots above shows the argument for a small circular contour γ centered around z = 0 for a function of the form f(z) = (z - 1)/z^n (which has a single pole of order n and no roots in γ) for n = 1, 2, and 3. Note that the complex argument must change continuously, so any "jumps" that occur as the contour crosses branch cuts must be taken into account.

complex argument | contour