Calculating Residues and using Cauchy Integral Formula

[Sekonda]

Hey,

I have a problem with this integral:

[tex]\int_{-\infty}^{\infty}dE\frac{1}{E^{2}-\mathbf{p}^{2}-m^{2}+i\epsilon}\: ,\: l^{2}=\mathbf{p}^{2}+m^{2}[/tex]

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

[tex]\int_{-\infty}^{\infty}dE\frac{1}{(E+(l-\frac{i\epsilon}{2l}))(E-(l-\frac{i\epsilon}{2l}))}[/tex]

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

[tex]-(l-\frac{i\epsilon}{2l})\: ,(l-\frac{i\epsilon}{2l})[/tex]

and we find the residues to be

[tex]-\frac{1}{2(l-\frac{i\epsilon}{2l})},\frac{1}{2(l-\frac{i\epsilon}{2l})}[/tex]

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

Any help is appreciated,
Thanks.

 

[DrDu]

You have to simplify the integrand using a partial fraction decomposition:

http://www.google.de/url?sa=t&rct=j...j4CgCw&usg=AFQjCNHZYQyU3q9P-4pho7uqGDx5OFOlpQ

 

[andrien]

you can take a large semicircle in either half planes.then go for it's evaluation.Only one residue will give contribution.