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How to use residues with improper integrals?

  1. May 22, 2007 #1
    1. The problem statement, all variables and given/known data

    integral^infty_0 1/(x^2+1) (I know the answer (from text) is pi/2)

    2. Relevant equations

    Well that is what I need help with. I can see that there are 2 roots to the x^2+1 that are +/- i and I know from the text that I use the arc in the where x >= 0 so I use the z=i residue. I'm not sure what to do next in order to get the answer.

    3. The attempt at a solution

    I know that it is an even function so I can multiply the answer by 1/2 to get the value, and I also understand that the value of the integral is 2*pi*i *[Sum of residues]

    Can anyone help me to understand. I have someone elses working out here on the same question and they are using limits and end up using ;im (z-i) = 1/2i
    z->i (z-i)(z+i)

    which I don't understand at all? How to get z-i and the other working out... it turns out that 1/2i is what i'm looking for because (1/2)2(pi)i(1/2i) is the answer.... can anyone give me any extra info that might help me to understand, after this question I have to do the same thing but the question is dx/(x^2+1)^2 so I'm hoping that if I get help with the first one I'll understand the 2nd...

    Thanks: :)
  2. jcsd
  3. May 22, 2007 #2


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    Science Advisor
    Homework Helper

    Yes. You need to use the residue from the pole at z=i. Since 1/(z^2+1)=1/((z+i)*(z-i)), the residue is the value of this function after dropping the (z-i) factor evaluated at z=i. 1/(z+i) at z=i is 1/(2i). This is a good easy example to be clear about before you move on to harder ones.
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