This image shows the real and imaginary parts of an oscillating function times the complex exponential oscillating at a different rate.
Thenub314
CC BY 3.0
<a href="//commons.wikimedia.org/wiki/File:Offfreq.svg" title="File:Offfreq.svg">File:Offfreq.svg</a>     Changes: 2πi is replaced by i2π, preferred by some.