# Finding the magnitude of a complex exponential function

1. Apr 20, 2013

### Malgrif

1. The problem statement, all variables and given/known data
I want to know the steps involved in finding the magnitude of a complex exponential function. An example of the following is shown in this picture:

2. Relevant equations
|a+jb|=sqrt(a^2+b^2)

|x/y|=|x|/|y|

3. The attempt at a solution

For the denominator, I replaced z with e^jw and used euler identity to expand the terms.

1-2rcos(w)e^(-jw)+r^2e^(-2jw)

1-rcos(w)*(cos(w)-jsinw)+.5r^2(cos(2w)-jsin(2w))

After simplifying I get:

1-rcos^2(w)+.5r^2cos(2w) + j(rsin(w)cos(w)-.5r^2sin(2w)

from there I let a=1-r(cos^2(w)-.5r^2cos(2w)) and b=r(sin(w)cos(w)-.5rsin(2w))

and using wolfram alpha to solve for sqrt(a^2+b^2) I don't get the simplified expression shown in the picture above. Am I approaching the problem correctly?

2. Apr 20, 2013

### Staff: Mentor

If z=e^(jw), you can use this directly in the first fraction, to get the factor of (1-r). The other factor is easy to handle then.

3. Apr 20, 2013

### Malgrif

Ah, I see. Thank you very much. I got the answer =)