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Math Help - complex analysis

  1. #1
    tescotime
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    complex analysis

    Hi, How would you show that for all z,w in the complex no. that ||z|-|w|| is less than or equal to |z-w| and that if the sequence z_n tends to w it implies that |z_n| tends to |w|.

    They probably follow a similar pattern to the proof for real numbers but I'm unsure about bits of it.

    It also asks how to show that for theta in the reals, ||z|-|w|| is less than or equal to |z+we^(itheta)| and hence that ||z|-|w|| = min{|z +we^(itheta)| : theta in reals}

    I'm not even sure if we've done this in class.

    thanks for any help
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  2. #2
    MHF Contributor
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    Quote Originally Posted by tescotime View Post
    How would you show that for all z,w in the complex no. that ||z|-|w|| is less than or equal to |z-w| and that if the sequence z_n tends to w it implies that |z_n| tends to |w|.
    Assuming that you know the triangle inequality |u+v|\leqslant|u|+|v|, all you have to do is to apply it with u=z-w and v=w. Then apply it again with u=w-z and v=z.

    The second part then follows easily from that result, because ||z_n|-|z||\leq|z_n-z|.
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