How to use residues with improper integrals?

[laura_a]

Homework Statement

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

Homework 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.

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: :)

 

[Dick]

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.