Let denote the change in the complex argument of a function around a contour . Also let denote the number of roots of in and denote the sum of the orders of all poles of lying inside . Then
(1)

For example, the plots above shows the argument for a small circular contour centered around for a function of the form (which has a single pole of order and no roots in ) for , 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.
To find in a given region , break into paths and find for each path. On a circular arc
(2)

let be a polynomial of degree . Then
(3)
 
(4)

Plugging in gives
(5)

So as ,
(6)

(7)

and
(8)

For a real segment ,
(9)

For an imaginary segment ,
(10)
