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Math Help - Exponential Inverse

  1. #1
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    Exponential Inverse

    Ok. I have to show that

    f(z)=exp(1-2iz)

    does not have an inverse function.

    Now I am trying to prove that the f(z) is not one one, but am having a hard time trying to come up with two values, that when inserted in f(z) result in the same answer.

    Any help would be great

    Thanx

    Bex
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting…

    s=f(z)=e^{1-2\cdot i\cdot z}

    … we derive…

    \ln s= 1-2\cdot i\cdot z \rightarrow z=\frac {1-\ln s}{2\cdot i}

    … which is the requested inverse function…

    Kind regards

    \chi \sigma
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  3. #3
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    Thanks but you have shown the inverse of f(z), however the question asks me to prove that f(z) DOES NOT have and inverse.



    Quote Originally Posted by chisigma View Post
    Setting…

    s=f(z)=e^{1-2\cdot i\cdot z}

    … we derive…

    \ln s= 1-2\cdot i\cdot z \rightarrow z=\frac {1-\ln s}{2\cdot i}

    … which is the requested inverse function…

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
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    mr fantastic's Avatar
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    Quote Originally Posted by bex23 View Post
    Ok. I have to show that

    f(z)=exp(1-2iz)

    does not have an inverse function.

    Now I am trying to prove that the f(z) is not one one, but am having a hard time trying to come up with two values, that when inserted in f(z) result in the same answer.

    Any help would be great

    Thanx

    Bex
    f(z) = f(z + n \pi) where n is an integer.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    f(z) = f(z + n \pi) where n is an integer.
    So would this give

    f(z+n \pi)=exp(1-2iz-2nipi)

    And because of the different values of n, f(z) would never be one one and thus have no inverse.
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