Integral to be solved

Question

I have been working on t-copulas and I have an integral that i need to solve. Could you help me on this please?

$\int_M^\infty e^{\lambda \epsilon}(1+\epsilon/\mu)^{-0.5(\mu+1)}d\epsilon$

Thank you

Answer 1

First of all, we must have $\lambda<0$ for it to converge.

Let $u=1+\epsilon/\mu$, assuming $\mu<0$,

$$=e^{-\lambda\mu}\int_{1+M/\mu}^{-\infty}e^{\lambda\mu u}u^{-0.5(\mu+1)}\ du$$

Now let $t=-\lambda\mu u$,

$$=\frac{(-\lambda\mu)^{1-0.5(\mu+1)}}{e^{\lambda\mu}}\int_{-\lambda(\mu+M)}^\infty e^{-t}t^{-0.5(\mu+1)}\ dt\\=\frac{(-\lambda\mu)^{1-0.5(\mu+1)}}{e^{\lambda\mu}}\Gamma(1-0.5(\mu+1),-\lambda(\mu+M))$$

where we use the upper incomplete gamma function.