Problems with identity in complex calc

[betel]

Hello,

in a paper I have the identity

$$\int_{-\infty}^{\infty} d x \sqrt{(x-i\epsilon)^2-1}= I_+ - I_- + i \int_{-1}^{1}(\ldots)$$


where $$I_+ = \int_1^{\infty}(\ldots), I_=\int_{-\infty}^{-1}(\ldots)$$ and $$\epsilon$$ is a small positive number that will be taken to zero at the end.

My Problem is to get the minus sign between $$I_+$$ and $$I_-$$. In all my calculations I get +. The integrand for both should be $$\sqrt{x^2-1}$$.

Can anybody tell me what I am missing here to get the correct sign.
Thanks in advance.
betel

 

[Landau]

I think it's essential what you (or the author) means by "(...)" for the integrand.

 

[betel]

In further calculations the author uses $$\sqrt{x^2-1}$$. The integrand is definitely real.