Complex exponential function

With n = ln a a real number

u = az = enz= e n(x + i y ) = e nx∙ e iny

= e nx (cos (ny) +i sin(ny))

Expon. growth function n= |n|  ➾ e z = e nx (cos (ny) +i sin(ny))

Expon. damping function n = -|n|  ➾ e -z= e - nx (cos (ny) - i sin(ny))

The length of the vector z is expanded exponentially by the real part of nz, its angle is rotated in the mathematical positive sense by n times the imaginary part of z

n=1 leads to the familiar exponential growth function

u = e x (cos (y) +i sin(y))