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.
© 1996-9 Eric W. Weisstein
1999-05-26