## Variation of Argument

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
 (3)

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.