Identifying Pole Order to the Test for Pole Procedure

NJunJie
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Homework Statement



Test for pole procedure.
As attached powerpoint.
Pse advise.

Homework Equations





The Attempt at a Solution

 

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Basically, what you have is right- although you should "suspect" that the pole at z= 7 is of order 3, not 7. I suspect that was a typo! A function, f(z) has "a pole of order n at z= z0" if and only if (z- z0)nf(z) has a non-zero limit as z goes to z0 and that is exactly what you have shown.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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