# Thread: What is the imaginary part?

1. ## What is the imaginary part?

What is the imaginary part of the complex number:

((z-1)/(z+1))^2; where z is an arbitrary complex number.

Please illustrate how you found it. It all seems a bit confusing to me (I tried multiplying the denominator by its conjugate - but it didn't seem to help; or I made a mistake while expanding)

Thanks!

2. Originally Posted by tombrownington
What is the imaginary part of the complex number:

((z-1)/(z+1)^2); where z is an arbitrary complex number.

Please illustrate how you found it. It all seems a bit confusing to me (I tried multiplying the denominator by its conjugate - but it didn't seem to help; or I made a mistake while expanding)

Thanks!
Substitute z = x + iy.

It's not difficult - it just requires care and perseverance (qualities I lack in situations like this).

Show your working and I'll critique any error.

3. Originally Posted by tombrownington
What is the imaginary part of the complex number:

((z-1)/(z+1)^2); where z is an arbitrary complex number.

Please illustrate how you found it. It all seems a bit confusing to me (I tried multiplying the denominator by its conjugate - but it didn't seem to help; or I made a mistake while expanding)

Thanks!
First we need to find the conjugate of $\displaystyle (z+1)^2$, which is $\displaystyle (\overline{z}+1)^2$
Then:

$\displaystyle \frac{z-1}{(z+1)^2}= \frac{(z-1)(\overline{z}+1)^2}{|(z+1)|^2}$ $\displaystyle =\frac{(z-1)(\overline{z}^2+2\overline{z}+1)}{|(z+1)|^2} =\frac{|z|^2\overline{z}+2|z|^2+z-\overline{z}^2-2\overline{z}-1}{|(z+1)|^2}$

Now expand the top using $\displaystyle z={\rm{re}}(z)+i ~{\rm{im}}(z)$ to find the imaginary part.

RonL

4. Hi Captain Black,
Thanks for the help, but you read my question when I had made a typographical error. The actual fraction who's imaginary part I wanted is:

$\displaystyle \frac {(z-1)^2}{(z+1)^2}$,

where z is an arbitrary complex number

5. Originally Posted by tombrownington
Hi Captain Black,
Thanks for the help, but you read my question when I had made a typographical error. The actual fraction who's imaginary part I wanted is:

$\displaystyle \frac {(z-1)^2}{(z+1)^2}$,

where z is an arbitrary complex number
Having seen what was done for the original post I'm sure you cn adapt that to this expression yourself.

RonL

6. Thanks Mr.Perfect & Captain Black,
I finally got the answer with a little bit of help from both of you.

7. Originally Posted by tombrownington
Hi Captain Black,
Thanks for the help, but you read my question when I had made a typographical error. The actual fraction who's imaginary part I wanted is:

$\displaystyle \frac {(z-1)^2}{(z+1)^2}$,

where z is an arbitrary complex number
Hi tombrwonington,

You have had some excellent replies already, but you might find it simpler to use the fact that the imaginary part of $\displaystyle w$ is $\displaystyle (1/2i) (w - \overline{w})$ and make the substitution $\displaystyle w = \frac {(z-1)^2}{(z+1)^2}$.

Just a suggestion...

jw