Functions in Two-Dimensional Vector Space or Complex Plane

move

m	1
n	6

a

b


axes

This Demonstration illustrates that functions in a real two-dimensional vector space are equivalent to functions in the field of complex numbers. Specifically, we consider a class of functions of the form

Z(z)=a

imt

e

+b

int

e

, where

m

and

n

are different integers. The black arrow traces out the function

Z(z)

, shown as a black curve, while the two component vectors are represented by red and blue arrows, tracing out red and blue circles.