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Math Help - Limit of complex function

  1. #1
    Newbie mukmar's Avatar
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    Limit of complex function

    Find  \displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}

    Attempt: I thought that this limit might not exist since if you let z = x + iy, then when approaching from y-axis, the limit equals iy and when approaching from the x-axis, the limit equals x.

    However the solution says that limit is actually 0. I'm not sure how to get this result.

    Any suggestions?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Use |\bar{z}^2/z|=|z|.

    Regards.
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  3. #3
    Newbie mukmar's Avatar
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    Is there a limit law that says that the limit of the modulus is related to the limit of the function in any way?
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by mukmar View Post
    Find  \displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}

    Attempt: I thought that this limit might not exist since if you let z = x + iy, then when approaching from y-axis, the limit equals iy and when approaching from the x-axis, the limit equals x.

    However the solution says that limit is actually 0. I'm not sure how to get this result.

    Any suggestions?
    Setting z=x + i\ y You obtain...

    \displaystyle \frac{\bar{z}^{2}}{z} = \frac{x^{2} - y^{2} -2 i x y}{x+i\ y}= \frac{x^{3} -3 x y^{2}}{x^{2}+y^{2}} + i\ \frac{y^{3} -3 x^{2} y}{x^{2}+y^{2}} (1)

    Now compute separately the \lim_{(x,y) \rightarrow (0,0) of the real and imaginary part of (1)...

    Kind regards

    \chi \sigma
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  5. #5
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    Quote Originally Posted by mukmar View Post
    Find  \displaystyle\lim_{z \to 0} \frac{\bar{z}^2}{z}

    Attempt: I thought that this limit might not exist since if you let z = x + iy, then when approaching from y-axis, the limit equals iy and when approaching from the x-axis, the limit equals x.

    However the solution says that limit is actually 0. I'm not sure how to get this result.

    Any suggestions?

    Another idea: write z in polar form, z=re^{i\phi}\Longrightarrow z\rightarrow 0\Longleftrightarrow r\rightarrow 0\,,\,\overline{z}=re^{-i\phi}, so:

    \displaystyle{\frac{\overline{z}^2}{z}=\frac{r}{e^  {i\phi}}\xrightarrow [r\to 0]{} 0}

    Tonio
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mukmar View Post
    Is there a limit law that says that the limit of the modulus is related to the limit of the function in any way?
    Better:

     |\bar{z}^2/z-0|<\epsilon\Leftrightarrow |z-0|<\epsilon\quad (z\neq 0)

    Now, choose \delta=\epsilon.

    Regards.
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