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This is my version of explanation. I would suggest you to come up with your own explanation. The best way would be for you to try explain this to somebody else in your own words.

Following is my version of explanation, but this is just an example. You may come up with a better version.

Exponential with Complex Number Exponent

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

tstep = 2*pi/40;

%t=0:tstep:tstep*40*2;

t=0:tstep:tstep*40*2;

k=1;

d=0;

e3 = exp(k*t*j)*exp(j*d);

hFig = figure(1);

subplot(2,2,2);

plot(t,real(e3));axis([0 4*pi -1.1 1.1]);

xlabel('t');

ylabel('real : e^{jkt},k=1.0');

grid on;

subplot(2,2,4);

plot(t,imag(e3));

axis([0 4*pi -1.1 1.1]);

xlabel('t');

ylabel('imaginary : e^{jkt},k=1.0');

grid on;

subplot(2,2,[1 3]);

plot(real(e3),imag(e3));

axis([-1.1 1.1 -1.1 1.1]);

xlabel('real : e^{jkt},k=1.0');

ylabel('imaginary : e^{jkt},k=1.0');

grid on;

pos = [(i/50.0) 0.01 (1.0/50.0) 0.98];

set(hFig,'Position',[300 400 800 300])