I am trying to solve this type of integral:
$$\int_{-\infty}^{\infty}\dfrac{e^{zt}}{(z^2-1)^2}dz.$$
I know that if the function in the integral has simple poles on the real axis then you can apply this formula:
$$\int_{-\infty}^{\infty}Q(z)dz=2 \pi j \sum Rez^+ +\pi j \sum Rez^0.$$
However, the function under my integral has poles of second order.
$z=1 ; z=-1$
What is the approach in this case? And how would different values for $t$ change the result?
For the Residue Theorem, the residue at $z_0$, which is a $m$-order singularity of $f(z)$, is:
$\frac{1}{\left(m-1\right)!} \lim_{z \to z_{0}} \frac{d^{m-1}}{dz^{m-1}} \{{\left(z-z_0\right)}^mf(z)\}$
Note: Alpha says this integral diverges.