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Math Help - Image of a function

  1. #1
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    Image of a function

    The question:

    Find the image of the following function:

    f(z)= z/(1-z)^2

    I solved this but I am not 100% sure.

    My solution

    if w=z/(1-z)^2 then for z!=1 then wz^2-(2w+1)z + w=0 and therefore

    z=(2w+1 + sqrt(4w+1))/2w or z=(2w+1 - sqrt(4w+1))/2w for every w!=0
    and for w=0 we can pick z=0.

    So for every w in C there exists a z in C so w=f(z) and therfore the image of f(z) is the entire complex plane C.

    I can't seem to contradict this, but still it dosent seem right to me...

    Would appreciate any comments

    SK
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    You should learn to trust your own solutions! If you've looked hard and you can't find a problem with it, at some point you should be confident that it's correct.

    The question you posted isn't even very precisely formulated anyways. What is the domain of the function? The Riemann sphere? The complex plane? Do you include the singularity z=1 in the domain? (If so, you have to include the point w=\infty in the image.)
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  3. #3
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    The domain was not specified. This is from a class in complex analysis, so the domain I beleive is, the largest subset of C. Therfore z!=1.
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