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Math Help - limits in complex plane

  1. #1
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    limits in complex plane

    If z_n = r_n e^{ix_n} converges to z = re^{ix}, one may ask why is it that x_n converges to x? One can show this directly using the fact that the (z_n) sequence converges iff its real and imaginary components converge. It is fairly obvious that r_n converges to r (use ||x|-|y||<=|x-y|). Using e^{ix} = cosx + isinx one can get an inequality involving sin(x_n), cos(x_n), cosx, and sinx, then an inequality in x_n,x alone that shows x_n converges to x.
    arlington
    Last edited by arlingtonbassett; January 28th 2013 at 04:16 PM.
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  2. #2
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    Re: limits in complex plane

    Hey arlingtonbassett.

    What kind of convergence are you talking about (point-wise for example)?
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  3. #3
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    Re: limits in complex plane

    I think you could make the argument directly. If z \ne 0, then a small enough \epsilon-ball around z will have a small range of \theta (where z=re^{i\theta}). Specifically, if |z_n-z|<\epsilon, then |\theta_n-\theta| < \frac{\epsilon}{r}.

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