How to integrate with branch cuts?

[krindik]

Hi,

I find integration with branch cuts difficult to grasp.
For example, I can understand that $$\sqrt{x}$$ is mutivalued and has 2 branches if we take a branch cut from 0 to +infty. But given it to be integrated from -infty to +infty what is the the meaning of taking a branch of $$\sqrt{x}$$ ?

Could u pls give me some pointers to look or guide me with this example?

I hope to find answers to integrals like $$\sqrt{(x-1){x-2}}$$

 

[maudape]

Did you get un answer for your question?

I'm dealing with this integral:
\int_0^{\infty } \frac{(\text{BesselJ}[0,\zeta a]-\text{BesselJ}[0,\zeta b])^2}{\zeta \sqrt{\zeta ^2-k^2}} \, d\zeta