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Questions on Complex Analysis

  • Thread starter raopeng
  • Start date
  • #1
86
0

Homework Statement


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.


Homework Equations


Ehh no idea what equations shall be used.


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.
 

Answers and Replies

  • #2
5,439
7
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.
 
  • #3
86
0
Thank you. Yes I noticed that too, as we can extract an infinite series from it...
 
  • #4
5,439
7
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)
 
  • #5
86
0
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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