How Can I Solve This Contour Integral with a Pole at Zero?

[Ana2015]

I want to solve this contour integral
$$J(\omega)= \frac{1}{2\pi}\frac{\gamma_i\lambda^2}{(\lambda^2+(\omega_i-\Delta-\omega)^2)} $$
$$N(\omega)=\frac{1}{e^{\frac{-\omega t}{T}}-1}$$

$$\int_0^\infty J(\omega)N(\omega)$$
there are three poles I don't know how I get rid of pole on zero (pole in N(w))
would you please help me?
Thanks

 

[mfb]

Doesn't look like a convergent integral to me.

For small ω,
$$N(\omega)=\frac{1}{e^{\frac{-\omega t}{T}}-1} \approx \frac{-T}{\omega t}$$

While J(ω) quickly approaches some constant.