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Math Help - Limits on Complex Numbers

  1. #1
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    Limits on Complex Numbers

    Here are two from my homework I can't quite figure out.

    1. \lim_{\Delta z \to 0} \frac {(z_0 + \Delta z )^3 - z_0^3}{\Delta z}

    2 \lim_{z \to 2\pi} (e^{iz} - e^{-iz})
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  2. #2
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    Quote Originally Posted by Creebe View Post
    Here are two from my homework I can't quite figure out.

    1. \lim_{\Delta z \to 0} \frac {(z_0 + \Delta z )^3 - z_0^3}{\Delta z}



    \lim_{\Delta_z \to 0} \frac {(z_0 + \Delta_z )^3 - z_0^3}{\Delta_z}=:f'(z_0)\,,\,with\,\,\,f(z)=z^3, or directly:

    \lim_{\Delta_z \to 0} \frac {(z_0 + \Delta_z )^3 - z_0^3}{\Delta_z}=\lim_{\Delta_z\to 0}\frac{3z_0^2\Delta_z+3z_0\Delta^2_z+\Delta^3_z}{  \Delta_z}=\lim_{\Delta_z\to 0}\left(3z_0^2+3z_0\Delta_z+\Delta^2_z\right)=3z_0  ^2



    2 \lim_{z \to 2\pi} (e^{iz} - e^{-iz})

    Hint: e^{iz}-e^{-iz}=2i\sin z\,,\,\,z\in\mathbb{C}

    Tonio
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