Complex Exponentials

[Travis Gibby]

I got Dr. Earl Geddes and Lidia Lee's book, "Audio Transducers" for christmas and I am already stuck on page 1. The topic is complex exponentials.

A complex exponential is an exponential raised to a complex power:

e^-iwt

where:

e^iz = cos(z)+i sin(z)

In this case the complex exponential represents a sine wave. Plotted on a polar graph it is a vector that rotates counter-clockwise at a rate of w radians per second. The frequency is w/2 Pi. t is time.

The thing that confuses me here is that the complex numbers that I am familiar with are in this form:

a + bi = r [cos (theta) + sin (theta)]

where:

r = (a^2 + b^2)^0.5

Theta is the angle of interest on a polar plot, and r is the magnitude of the vector.

I don't know how to plot a function in this form:

e^-iwt

Can anyone help?

Travis

 

[poobah]

Hmmmm...

Use w (omega) as your "x" axis... should be a decaying sinusoid...

Is this what you are asking?

 

[omba]

e^-iwt = e^i(-wt)

so,
-wt = theta
in your expression

a+bi = r(cos(-wt) + i*sin(-wt))

Will this help?

 

[joensd]

I don't know how to plot a function in this form:

You gave the answer yourself:

In this case the complex exponential represents a sine wave. Plotted on a polar graph it is a vector that rotates counter-clockwise at a rate of w radians per second. The frequency is w/2 Pi. t is time.

It´s a rotating pointer that has an amplitude of 1.
It starts in the direction of the "real"-axis and represents a sine-wave.
wikipedia

 

[Travis Gibby]

Thanks everyone for the help. I made a short program to plot this function in time based on your advice. It works just as the book says, a vector rotating at omega radians/sec .

Thanks again,

Travis