I plotted the real and imaginary parts of the complex exponential $e^{j\pi n}$ in MATLAB and got an unexpected result for the complex portion as well as something similar for $2\pi$. I was wondering why it appears as it does because I thought it would just oscillate between $\pm1$. I also plotted $e^{j(\pi/2)n}$ as well as $e^{j(3\pi/2)n}$ and both have a complex part that oscillates between $\pm1$.
Here is the code so you might reproduce it.
% Frequency
W0 = pi;
% Time
n = -20:20;
% Signal
j = sqrt(-1);
x = exp(j*W0*n);
% Plotting commands
figure(1)
subplot(2,1,1), stem(n,real(x)), xlabel('n'), title('Real part of e^{jW_0 n}');
subplot(2,1,2), stem(n,imag(x)), xlabel('n'), title('Imag. part of e^{jW_0 n}');
The real part of $e^{\pi n i}$ alternates between $-1$ and $1$.
The imaginary part is nearly zero on the plot (notice $10^{-14}$ on the vertical axis). Theoretically, it should be exactly zero, but the computations introduce some small errors (rounding, truncation of power series, etc.)