The argument principle

Draw a closed contour in the left plane. Its image f(contour) in the right plane winds around the origin once for every zero minus every pole of f it encloses. Hover to see the image of a small circle around the cursor.

A modern JavaScript + Canvas port of a 2000 Java applet by Terence Tao (originally ArgumentApp.java / ArgumentGrid.java).

f(z) = ( z² + z + ) / ( z² + z + )

z-plane — draw a contourf(z)-plane (winds around 0)

Draw a loop; the winding number = (zeros − poles) enclosed.