Counting the Number of Roots of Transcendental Functions in Bounded Regions Using Winding Numbers

function

3z



+20cos(z)-1

radius

1

zoom

50

+

For a given analytic function

f

chosen from the popup menu, a circle radius

r

chosen using a slider, and a circle center denoted by "+" (which can be dragged), the graphic on the left shows the roots of

f

inside the disk of radius

r

. The graphic on the right shows the image of the circle under the map

f

and the origin of the coordinate system. According to the argument principle, the number of roots of

f

(counted with multiplicities) inside the disk is equal to the winding number of the image of the circle around the origin.

When the circle is dragged, its image under

f

can become very large or very small. A zoom control is provided that can be used to see the entire closed curve or, alternatively, the behavior of the curve near the origin (the little red point in the graphic on the right). Zooming in and out may be necessary to correctly count the winding number.