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Can someone explain to me how this is not 1/4? Complex Analy

  1. Aug 6, 2015 #1

    RJLiberator

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    Gold Member

    1. The problem statement, all variables and given/known data
    Hi all, I posted the image of the solution here.

    My questions concerns the evaluation of the Residue at z=i.

    Screen Shot 2015-08-06 at 10.40.56 PM.png

    2. Relevant equations
    None needed, all giving in the question -- simple algebraic mistake made is likely to be the problem here.

    3. The attempt at a solution


    We see that phi function is clearly equal to 1/(z+i)^2 as the show.
    Since it is a pole of order two, we must differentiate this function first before evaluation at singular point i.

    The derivative of this function should be -2/(z+i)^(3)

    Now we can evaluate this at point z=i.
    -2/(2i)^3 = -2/8i^3 = -1/4i^3. When I calculate this, this becomes i/4

    But the answer states that this is instead 1/4i.

    Where am I going wrong in my complex algebra?
     
    Last edited: Aug 6, 2015
  2. jcsd
  3. Aug 6, 2015 #2
    Looks like a simple algebra mistake. When you simplify for the second time you eliminate the negative sign, effectively dividing by -1. That's why your answer differs by a factor of -1.
     
  4. Aug 6, 2015 #3

    RJLiberator

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    Gold Member

    Wopps. good catch, that was a mistake in my post, but not the mistake i wanted to clarify. (I've seen edited my original post for future readers).

    The problem is, when I evaluate this, it comes to -1/4i^3 and I believe this is equal to i/4 as per wolfram alpha, etc.
    However, in the image, it shows that this is not the case. Instead, the image shows the solution to be 1/4i.

    What's going on here? My only other line of thinking is this: They split up -1/4i^3 into -1/(4i^2*i)

    Oh...

    That works...

    -1/(4i^2*i) = 1/4i
    But, why would they not simplify this further so that it is i/4? The final answer would then be the negative of what the solution offers.
     
  5. Aug 6, 2015 #4
    But, why would they not simplify this further so that it is i/4?
    Because that would not be a valid simplification. 1/i = -i, and so we would need -i/4 and not i/4.

    I'd be interested in seeing this Wolfram Alpha output you speak of ...
     
  6. Aug 6, 2015 #5

    RJLiberator

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    Gold Member

    Ah, I see. The that is indeed the mistake I was performing. Did not have parentheses around the denominator in wolfram alpha.

    So it does come out to be -i/4. Which with that negative factor, makes them equivalent.

    You have successfully helped me solve this problem and furthered my understanding.
    Thank you.
     
  7. Aug 6, 2015 #6
    Aha! That's the reason.

    Glad I could help.
     
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