Rouche's Theorem
Suppose

A functionis analytic on a simply connected region

is a simple closed contour in

is analytic onandfor
Thenhas the same number of zeros asinsidecounted according to their multiplicity.
Proof: By 2 above, ifthenlies inside the open disc with centreand radius
Sincecannot pass on the opposite side of the origin asso andcircle the origin the same number of times and)
Example: Takeandand letbe the circletraversed anticlockwise.
Thenandonso all the conditions of Rouche's theorem are satisfied andandboth have winding number three about the origin.