Arcs and rays (centered at -n) under Pn(z) = (1+z/n)n

image of arcs and rays under (1+z/n)^n

image of arcs and rays under (1+z/n)^n

These diagrams were done to illustrate Needham, Chapter 2, Exercise 22(ii). Pn translates the center of the plot back to the origin. With increasing n,

  • Pn(z) contracts the dimensions, as can be seen by comparing the axis ticks on the right hand side of both images. Two of the arcs have been reduced to a point at the origin.
  • The velocity of winding of Pn(z) increases with the power mapping, resulting in larger angles between adjacent rays.
  • The segments of the arcs begin to be distinguishable due to the increasing velocity of the winding.
November 17, 2014, Gary B. Palmer