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Math Help - A Limit Problem

  1. #1
    Senior Member bkarpuz's Avatar
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    A Limit Problem

    Dear friends,

    I need help in showing the following.
    \lim_{\substack{\lambda\in\mathbb{R}\\ \lambda\to0}}\bigg(\frac{1}{\lambda}\log|1+z\lambd  a|\bigg)=\mathrm{Re}(z)

    Thanks!

    bkarpuz
    Last edited by bkarpuz; February 9th 2010 at 11:56 AM.
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  2. #2
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    Quote Originally Posted by bkarpuz View Post
    Dear friends,

    I need help in showing the following.
    \lim_{\lambda\to0}\bigg(\frac{1}{\lambda}\log|1+z\  lambda|\bigg)=\mathrm{Re}(z)

    Thanks!

    bkarpuz
    (you should specify that \lambda\in\mathbb{R})

    After expanding you get |1+z\lambda|^2=1+2{\rm Re}(z)\lambda+o(\lambda) when \lambda\to 0, \lambda\in\mathbb{R}, from which the result follows quickly using \log (1+u)=u+o(u) when u\to 0, and \log u=\frac{1}{2}\log u^2.
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  3. #3
    Senior Member bkarpuz's Avatar
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    Angry

    Quote Originally Posted by Laurent View Post
    (you should specify that \lambda\in\mathbb{R})

    After expanding you get |1+z\lambda|^2=1+2{\rm Re}(z)\lambda+o(\lambda) when \lambda\to 0, \lambda\in\mathbb{R}, from which the result follows quickly using \log (1+u)=u+o(u) when u\to 0, and \log u=\frac{1}{2}\log u^2.
    How did I miss this?! :S
    Thanks for your reply Laurent, it has been a long time not heard from you. :]

    Actually I am working with the quotient when \lambda\in\mathbb{C} and its making me confused! :S
    Last edited by bkarpuz; February 9th 2010 at 12:18 PM.
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  4. #4
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by bkarpuz View Post
    How did I miss this?! :S
    Thanks for your reply Laurent, it has been a long time not heard from you. :]

    Actually I am working with the quotient when \lambda\in\mathbb{C} and its making me confused! :S
    Actually, the limit can be computed when \mathrm{Re}(\lambda)\neq0.
    In this case, we have \lambda=r\mathrm{e}^{i\theta} with \theta\neq\pm\pi/2.
    So that, for the function f(z):=|z|^{-1}\log|1+z| for z\in\mathbb{C}\backslash\{-1\}, we have
    \lim_{r\to0^{+}}f(r\mathrm{e}^{i\theta})=\lim_{r\t  o0^{+}}\frac{1}{2r}\log\big(1+2r\cos(\theta)+r^{2}  \big)=\cos(\theta)
    by using the fact mentioned previously by Laurent ( \lim\nolimits_{\lambda\in\mathbb{R},\ \lambda\to0}f(\lambda)=1).
    On the other hand if \mathrm{Re}(\lambda)=0, it can be easily computed as
    \lim_{r\to0^{+}}f(r\mathrm{e}^{\pm i\pi/2})=\lim_{r\to0^{+}}\frac{1}{2r}\log\big(1+r^{2}\b  ig)=0.
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