Help with graphing an exponential function

Question

I know this is probably a very easy question, but how do you know the difference between an exponential that is decaying at an increasing or decreasing rate? I multiplied $e^{i2\omega}$ by a box and got kind of a semi circle where the rate of decay increased, while I expected it to decay at a decreasing rate on both sides.

Here was my mathematica input:

E^(I 2 ω) (HeavisideTheta[ω + Pi/4] - HeavisideTheta[ω - Pi/4])

Answer 1

The real part of $e^{2i\omega}$ is $\cos 2\omega,$ which in the interval that you defined resembles part of a circle.