The argument principle relates the numbers of zeros and poles of a meromorphic
function to a contour integral of its logarithmic derivative. Let 
be a counterclockwise simple closed contour
and let 
be meromorphic on an open region
containing 
and its interior, with no zeros or poles
on 
.
If 
and 
are the numbers of enclosed zeros and poles,
counted according to multiplicity and order, then
 |
(1)
|
The complex residue of 
at a root is its multiplicity,
and at a pole it is the negative order. The formula follows
from the residue theorem (Orloff 2018, pp. 1-2).
Let 
be the image contour, with its traversal retained. Setting
gives
 |
(2)
|
Thus,
 |
(3)
|
where 
is the contour winding number about the
origin and 
is the net variation
of argument along 
. The image contour need not
be simple, and tracing the same curve repeatedly contributes
with its traversal multiplicity.
For example, 
has three zeros counted with multiplicity
and one pole inside the unit
circle. Its logarithmic derivative
is 
,
so the image of the counterclockwise unit circle has
contour winding number 2 about the origin.
