# Complex Number- Express in magnitude/phase form

1. Sep 8, 2012

### aqmal_12

Hi, I have a problem with complex number. I do really appreciate your help. I've attempted the question but it's getting me no where. Thanks in advance!

1. The problem statement, all variables and given/known data

Perform the following complex variable calculations, using complex exponentials. Express the results in magnitude/phase form

a/(iw+a)

2. Relevant equations

3. The attempt at a solution

I just attempted this solution but it's getting me nowhere.

a/(a+iw) x (a-iw)/(a-iw) = a(a-iw)/a^2 + w^2

and as I factorize it further, I get the initial problem back. How do I transform this to the form of magnitude and phase? thank you so much.

2. Sep 9, 2012

### HallsofIvy

Staff Emeritus
First, you need parentheses- you mean $a(a- iw)/(a^2+ w^2)= (a^2/(a^2+w^2))- i (aw/(a^2+ w^2)$. Frankly, the difficulty appears toijn be that the problem asks you to "use complex exponentials", writing the "magnitude" and "phase" and you don't seem to know what those are! This problem expects you to know that the "complex exponential" for x+ iy is $(\sqrt{x^2+ y^3})e^{i arctan(y/x)}$. Just do that with x= a^2/(a^2+ w^2) and y= aw/(a^2+ w^2).

But the problem says to do the calculations "using complex exponentials" and I would interpret that as meaning to change into complex exponentials and then do the caculation. The caculation is a/(a+ iw). The "complex exponential" for a is easy- a is a real number so its argument is 0 or $\pi$ and its distance from the origin is |a|- its complex exponential is $a e^{i0}$ if a is positive and $-a e^{i\pi}$ if a is negative. And the complex exponential of a+ iw is just what I showed above: $\sqrt{a^2+ w^2}e^{i arctan(w/a)}$.