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Thread: Proving the complex Log function is not continuous

  1. #1
    Junior Member CuriosityCabinet's Avatar
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    Proving the complex Log function is not continuous

    The question is:

    Let $\displaystyle Log z = ln \left | z \right | + i \theta $ where $\displaystyle -\pi < \theta \leq \pi $and $\displaystyle z=\left | z \right | e^{i\theta}$, $\displaystyle (z\neq 0)$. Prove that Log is not continuous on$\displaystyle (- \infty,0).$

    Hint: Consider the sequences {-1+i/n} and {-1-i/n}.

    I am not sure how I should be using these sequences in the proof.

    Please help!
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Proving the complex Log function is not continuous

    There's something alluring about giving late replies.



    Let's see what we have.
    -The function $\displaystyle f(z)=Log(z), z\in \mathbb{C}-(-\infty,0)$
    -The point -1, which belongs to the interval $\displaystyle (-\infty,0)$
    -Two sequences, $\displaystyle x_n=-1+i/n, y_n=-1-i/n, n\in \mathbb{N}$, which converge to -1.

    If the function were continuous at -1, then the limits $\displaystyle \lim f(x_n), \lim f(y_n)$ should be equal.

    But...
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