Determining Residues and using Cauchy Integral Formula

Hey,

I have a problem with this integral:

$\displaystyle \int_{-\infty}^{\infty}dE\frac{1}{E^{2}-\mathbf{p}^{2}-m^{2}+i\epsilon}\: ,\: l^{2}=\mathbf{p}^{2}+m^{2}$

The integration over all energies (arising in the loop function for calculating the scattering), I understand we write the above in this form:

$\displaystyle \int_{-\infty}^{\infty}dE\frac{1}{(E+(l-\frac{i\epsilon}{2l}))(E-(l-\frac{i\epsilon}{2l}))}$

Where ε is small and so the factor arising from it multiplying by itself can be neglected. It seems to evaluate this we can either calculate the residues of the two poles and sum them up and multiply by 2pi*i or we can use Cauchy Integral's formula - though I think it's the same thing... not really sure.

Our poles are at

$\displaystyle -(l-\frac{i\epsilon}{2l})\: ,(l-\frac{i\epsilon}{2l})$

and we find the residues to be

$\displaystyle -\frac{1}{2(l-\frac{i\epsilon}{2l})},\frac{1}{2(l-\frac{i\epsilon}{2l})}$

But I'm not sure how we see this or do this exactly...

Any help is appreciated,

Thanks.

Re: Determining Residues and using Cauchy Integral Formula

Hey Sekonda.

Did you try expanding out 1/(z-a) * 1/(z-b) into partial fractions and then using the results for residues?

Re: Determining Residues and using Cauchy Integral Formula

Indeed I did just last night, but I'm glad you've confirmed what I've done is correct. Still confused at why the 1/z contribution is the one we want, will need to look this up - unless someone can explain!

Thanks chiro,

SK