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Math Help - What is the imaginary part?

  1. #1
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    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!
    Last edited by tombrownington; May 3rd 2008 at 09:55 PM. Reason: typographical error
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  2. #2
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    Quote Originally Posted by tombrownington View Post
    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.
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    Quote Originally Posted by tombrownington View Post
    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 (z+1)^2, which is (\overline{z}+1)^2
    Then:

    \frac{z-1}{(z+1)^2}= \frac{(z-1)(\overline{z}+1)^2}{|(z+1)|^2} =\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 z={\rm{re}}(z)+i ~{\rm{im}}(z) to find the imaginary part.

    RonL
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    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:

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

    where z is an arbitrary complex number
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    Quote Originally Posted by tombrownington View Post
    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:

    \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
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  6. #6
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    Thanks Mr.Perfect & Captain Black,
    I finally got the answer with a little bit of help from both of you.
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    Quote Originally Posted by tombrownington View Post
    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:

    \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 w is (1/2i) (w - \overline{w}) and make the substitution w = \frac {(z-1)^2}{(z+1)^2} .

    Just a suggestion...

    jw
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