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Questions on Complex Analysis

  1. Sep 27, 2012 #1
    1. The problem statement, all variables and given/known data
    To prove that the sequence [itex]a_{n}= \prod_{k}^\infty (1 + \frac{i}{k})[/itex] when n is infinite constitutes points on a circle.


    2. Relevant equations
    Ehh no idea what equations shall be used.


    3. The attempt at a solution
    A friend asked me this, but I am usually engaged more with the physical aspects of Complex Analysis... so I have no idea how I should approach this question.
     
  2. jcsd
  3. Sep 27, 2012 #2
    Well points on a circle have a particular (finite) radius.

    So you need to prove your infinite product has a finite limit,
    Since we are in the complex plane this will be a radius, as every point with this modulus will be included.
     
  4. Sep 28, 2012 #3
    Thank you. Yes I noticed that too, as we can extract an infinite series from it...
     
  5. Sep 28, 2012 #4
    So what happens if you write a few of the terms of the series out and multiply them in pairs?

    Further hint put 1/k = α.
    So the terms take the form (1+αi)
     
  6. Sep 28, 2012 #5
    Thank you for the help. Also [itex]arg a_{n} = \sum^{\inf}_{k}\frac{1}{k}[/itex] covers the entire circle.
     
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