Let denote the change in argument of a function around a closed loop . Also let denote
the number of Roots of in and denote the number of Poles of in
. Then

(1) 
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
Plugging in
gives

(4) 

(5) 
so

(6) 
and

(7) 
For a Real segment ,

(8) 
For an Imaginary segment ,

(9) 
Note that the Argument must change continuously, so ``jumps'' occur across inverse tangent asymptotes.
© 19969 Eric W. Weisstein
19990526