# Complex integral with singularity

1. Mar 14, 2008

### blue2script

Hi all!

I am not so deep into complex analysis. So, for the following integral

$$$e^{ - \frac{1} {4}\int {\partial _{\bar x} \ln \left( {\left| \omega \right|^2 + 1} \right)d\bar x} }$$$

(where $$\bar x = x_0 - i x_1$$)I did some naive

$$$e^{ - \frac{1} {4}\int {\partial _{\bar x} \ln \left( {\left| \omega \right|^2 + 1} \right)d\bar x} } = e^{ - \frac{1} {4}\left. {\ln \left( {\left| \omega \right|^2 + 1} \right)} \right|_0^{\bar x} }$$$

Because the (analytic) function $$\omega$$ gets infinity for x = 0, the result doesn't make sense. I guess I failed in handling the singularities. Moreover I don't know on which line to integrate in the complex plane - from the problem itself there is no outstanding one.

Hope somebody can give me some hints!

A big thanks in advance, its really important!!

Blue2script