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Math Help - Limit of a complex function with cubed conjugate

  1. #1
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    Limit of a complex function with cubed conjugate

    Hello everyone,

    I'm working a problem set on complex analysis and encountered this limit, which I'm not sure how to solve. This is what I did. Limit of a complex function with cubed conjugate-img102.jpg

    Can anybody tell me wether it is correct or how to compute it properly?

    Thank you very much!
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  2. #2
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    Re: Limit of a complex function with cubed conjugate

    this works. You can save a few steps by noting that

    $|\bar{z}|=|z|$
    Thanks from Phyba and HallsofIvy
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    Re: Limit of a complex function with cubed conjugate

    Thank you very much, Romsek!
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    Re: Limit of a complex function with cubed conjugate

    You might also consider using polar form.

    If $z=r\exp(i\theta)$ then $\overline{~z~}^3=r^3\exp(-3i\theta)$

    Form which you get $\dfrac{\overline{~z~}^3}{z}=r^2\exp(-4i\theta)$ where $\theta\to 0~\&~r\to 0$
    Thanks from Phyba
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    Re: Limit of a complex function with cubed conjugate

    That seems more elegant, thanks Plato
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    Re: Limit of a complex function with cubed conjugate

    So when solving a complex limit with z-->0 I can write in polar form and make both z and r go to zero? I wasn't sure if the argument goes necessarily to zero when z does or only the modulus
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    Re: Limit of a complex function with cubed conjugate

    Quote Originally Posted by Phyba View Post
    So when solving a complex limit with z-->0 I can write in polar form and make both z and r go to zero? I wasn't sure if the argument goes necessarily to zero when z does or only the modulus
    As $z \to 0$ the modulus goes to zero while the exponential part spins around the unit circle with magnitude 1. So the overall magnitude goes to 0.
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    Re: Limit of a complex function with cubed conjugate

    So that means that arg(z) doesn't go to zero when z goes to zero, right? (I meant to say both theta and r earlier, I'm sorry)
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    Re: Limit of a complex function with cubed conjugate

    Quote Originally Posted by Phyba View Post
    So that means that arg(z) doesn't go to zero when z goes to zero, right? (I meant to say both theta and r earlier, I'm sorry)
    in this case no. You can contrive things so both the modulus and the argument go to zero but in this case only the modulus does. Note that speaking of the argument of 0 is nonsensical as it can be any value.
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    Re: Limit of a complex function with cubed conjugate

    mm.. I see. Thanks!
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