Complex Sine Waves

The main thing to understand here is that the complex sine wave takes on a simpler form due to to Eulers Equation, and that plotting it over time gives a 3D result.

You can take a sine wave and make it negative, simply by multiplying its formula with i. To make a complex sine wave, you can have a real sine wave added to an imaginary one, but we should pick different phases for each. The phases that make the most sense would be 0, and π/2, as they would represent the rotation from the the origin along the complex plane, and point to the real and imaginary axis respectfully.

Remember however, cosine is just sine already phase offset in a different direction. 

So if we go the opposite direction by flipping cosine and sine around such that we dont have to write out the phase offsets of each, we actually end up back to the right hand sides of Euler's equation.

Here we can set k to the full "2πft + θ" , which can also be substituted to the right side of the Euler equation as well, leaving us with:

This is shorthand for an equation of a complex sine wave. But what does this mean?

Remember a real sine wave goes along time (x) with the y value being the result of the of the some wave, that result being a real number on the number line. Ergo a real sine wave plotted over time outputs a 3D graph. However, a complex number is a number across the complex plane, not the number line, and a result of a complex sine wave given a time is a result on the 2D complex plane. Ergo, plotting a complex sine wave results in a 3D output. Given the phase offset inherit to the real and imaginary components, this results in the complex output looking like a corkscrew. If you looked at the real and imaginary parts separately, each would be a 2D output.

Make sure you understand how the complex sine wave becomes 3D over time before continuing.