I'm trying to calculate this integral :
$$I(z,k,a)= \int_1^\infty t^2 \operatorname{ArcTanh} \left(\sqrt{\frac{t^2-1}{t^2}} \dfrac{k}{z}\right)\, e^{-a\,t} \, dt$$
Where :
$\operatorname{ArcTanh(\cdots)}$ : is the inverse hyperbolic tangent function.
$z=x+i\,y$ : is a complex with the assumption $y\ll x$
$k$ and $a$ are reals $> 0$
Please, how do I go about this ? Is there any tricks that can be applied ?
Is there a way to transform this integral in a form of Plemelj-Dirac Formula ?
$$ I(z,k,a)=P \int_1^{t_0} \cdots\cdots \, dt \pm i \pi \big( \cdots \cdots \big)_{t=t_0}$$
Thanking you in advance.