# Integral of a multivalued function

I am trying to calculate an integral of a multivalued function, which has the form: \begin{align} I=&\int_0^{\infty} d\tau \ \left( \frac{1}{v_c \tau+\alpha} \right)^{\frac{1}{4}(K_c+1/K_c)} \left( \frac{1}{v_s \tau+\alpha} \right)^{\frac{1}{4}(K_s+1/K_s)} \left[ \frac{1}{(v_c \tau+\alpha)^2+4x^2} \right]^{-\frac{1}{8}(K_c-1/K_c)} \\ &\times \left[ \frac{1}{(v_s \tau+\alpha)^2+4x^2} \right]^{-\frac{1}{8}(K_s-1/K_s)} e^{-\frac{\pi}{4L}(v_{c,N}+v_{s,N})\tau}e^{i\omega\tau} \end{align}

here all the parameters ($$K_{c/s}, v_{c/s}, v_{c/s,N}, x, \omega, L$$) are positive and the $$\alpha$$ is an infinitesimal positive number. I tried to use Mathematica, but it doesn't give me any result. Perhaps this can be done within the context of complex integral.

Hope someone has more experience can give some suggestions, thanks in advance!

• About the parameters, $v_c$ and $v_s$ can be equals or are diferents? The same question about $K_s$ and $K_c$. – popi Jun 18 at 8:39
• @popi They can be equal, and here I would like to know the result for a general situation. – Chuan Chen Jun 18 at 12:37